id int32 | title string | problem string | question_latex string | question_html string | numerical_answer string | pub_date string | solved_by string | diff_rate string | difficulty string |
|---|---|---|---|---|---|---|---|---|---|
768 | Chandelier | A certain type of chandelier contains a circular ring of $n$ evenly spaced candleholders.
If only one candle is fitted, then the chandelier will be imbalanced. However, if a second identical candle is placed in the opposite candleholder (assuming $n$ is even) then perfect balance will be achieved and the chandelier wil... | A certain type of chandelier contains a circular ring of $n$ evenly spaced candleholders.
If only one candle is fitted, then the chandelier will be imbalanced. However, if a second identical candle is placed in the opposite candleholder (assuming $n$ is even) then perfect balance will be achieved and the chandelier wil... | <p>A certain type of chandelier contains a circular ring of $n$ evenly spaced candleholders.<br>
If only one candle is fitted, then the chandelier will be imbalanced. However, if a second identical candle is placed in the opposite candleholder (assuming $n$ is even) then perfect balance will be achieved and the chandel... | 14655308696436060 | Sunday, 17th October 2021, 05:00 am | 202 | 95% | hard |
36 | Double-base Palindromes | The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$.
(Please note that the palindromic number, in either base, may not include leading zeros.) | The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases.
Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$.
(Please note that the palindromic number, in either base, may not include leading zeros.) | <p>The decimal number, $585 = 1001001001_2$ (binary), is palindromic in both bases.</p>
<p>Find the sum of all numbers, less than one million, which are palindromic in base $10$ and base $2$.</p>
<p class="smaller">(Please note that the palindromic number, in either base, may not include leading zeros.)</p> | 872187 | Friday, 31st January 2003, 06:00 pm | 96478 | 5% | easy |
846 | Magic Bracelets | A bracelet is made by connecting at least three numbered beads in a circle. Each bead can only display $1$, $2$, or any number of the form $p^k$ or $2p^k$ for odd prime $p$.
In addition a magic bracelet must satisfy the following two conditions:
no two beads display the same number
the product of the numbers of any... | A bracelet is made by connecting at least three numbered beads in a circle. Each bead can only display $1$, $2$, or any number of the form $p^k$ or $2p^k$ for odd prime $p$.
In addition a magic bracelet must satisfy the following two conditions:
no two beads display the same number
the product of the numbers of any... | <p>
A <i>bracelet</i> is made by connecting at least three numbered beads in a circle. Each bead can only display $1$, $2$, or any number of the form $p^k$ or $2p^k$ for odd prime $p$.</p>
<p>
In addition a <i>magic bracelet</i> must satisfy the following two conditions:</p>
<ul>
<li> no two beads display the same numb... | 9851175623 | Saturday, 3rd June 2023, 08:00 pm | 224 | 50% | medium |
926 | Total Roundness | A round number is a number that ends with one or more zeros in a given base.
Let us define the roundness of a number $n$ in base $b$ as the number of zeros at the end of the base $b$ representation of $n$.
For example, $20$ has roundness $2$ in base $2$, because the base $2$ representation of $20$ is $10100$, which en... | A round number is a number that ends with one or more zeros in a given base.
Let us define the roundness of a number $n$ in base $b$ as the number of zeros at the end of the base $b$ representation of $n$.
For example, $20$ has roundness $2$ in base $2$, because the base $2$ representation of $20$ is $10100$, which en... | <p>
A <strong>round number</strong> is a number that ends with one or more zeros in a given base.</p>
<p>
Let us define the <dfn>roundness</dfn> of a number $n$ in base $b$ as the number of zeros at the end of the base $b$ representation of $n$.<br/>
For example, $20$ has roundness $2$ in base $2$, because the base $2$... | 40410219 | Saturday, 4th January 2025, 10:00 pm | 533 | 10% | easy |
46 | Goldbach's Other Conjecture | It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
\begin{align}
9 = 7 + 2 \times 1^2\\
15 = 7 + 2 \times 2^2\\
21 = 3 + 2 \times 3^2\\
25 = 7 + 2 \times 3^2\\
27 = 19 + 2 \times 2^2\\
33 = 31 + 2 \times 1^2
\end{align}
It turns out that the co... | It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.
\begin{align}
9 = 7 + 2 \times 1^2\\
15 = 7 + 2 \times 2^2\\
21 = 3 + 2 \times 3^2\\
25 = 7 + 2 \times 3^2\\
27 = 19 + 2 \times 2^2\\
33 = 31 + 2 \times 1^2
\end{align}
It turns out that the co... | <p>It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.</p>
\begin{align}
9 = 7 + 2 \times 1^2\\
15 = 7 + 2 \times 2^2\\
21 = 3 + 2 \times 3^2\\
25 = 7 + 2 \times 3^2\\
27 = 19 + 2 \times 2^2\\
33 = 31 + 2 \times 1^2
\end{align}
<p>It turns out t... | 5777 | Friday, 20th June 2003, 06:00 pm | 67702 | 5% | easy |
580 | Squarefree Hilbert Numbers | A Hilbert number is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a squarefree Hilbert number as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times13$. However $6237$ is a Hi... | A Hilbert number is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a squarefree Hilbert number as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times13$. However $6237$ is a Hi... | <p>
A <strong>Hilbert number</strong> is any positive integer of the form $4k+1$ for integer $k\geq 0$. We shall define a <i>squarefree Hilbert number</i> as a Hilbert number which is not divisible by the square of any Hilbert number other than one. For example, $117$ is a squarefree Hilbert number, equaling $9\times... | 2327213148095366 | Sunday, 4th December 2016, 04:00 am | 282 | 75% | hard |
449 | Chocolate Covered Candy | Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation:
$b^2 x^2 + b^2 y^2 + a^2 z^2 = a^2 b^2$.
Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter ... | Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation:
$b^2 x^2 + b^2 y^2 + a^2 z^2 = a^2 b^2$.
Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one millimeter ... | <p>Phil the confectioner is making a new batch of chocolate covered candy. Each candy centre is shaped like an ellipsoid of revolution defined by the equation:
$b^2 x^2 + b^2 y^2 + a^2 z^2 = a^2 b^2$.</p>
<p>Phil wants to know how much chocolate is needed to cover one candy centre with a uniform coat of chocolate one m... | 103.37870096 | Sunday, 8th December 2013, 04:00 am | 1004 | 40% | medium |
254 | Sums of Digit Factorials | Define $f(n)$ as the sum of the factorials of the digits of $n$. For example, $f(342) = 3! + 4! + 2! = 32$.
Define $sf(n)$ as the sum of the digits of $f(n)$. So $sf(342) = 3 + 2 = 5$.
Define $g(i)$ to be the smallest positive integer $n$ such that $sf(n) = i$. Though $sf(342)$ is $5$, $sf(25)$ is also $5$, and it can ... | Define $f(n)$ as the sum of the factorials of the digits of $n$. For example, $f(342) = 3! + 4! + 2! = 32$.
Define $sf(n)$ as the sum of the digits of $f(n)$. So $sf(342) = 3 + 2 = 5$.
Define $g(i)$ to be the smallest positive integer $n$ such that $sf(n) = i$. Though $sf(342)$ is $5$, $sf(25)$ is also $5$, and it can ... | <p>Define $f(n)$ as the sum of the factorials of the digits of $n$. For example, $f(342) = 3! + 4! + 2! = 32$.</p>
<p>Define $sf(n)$ as the sum of the digits of $f(n)$. So $sf(342) = 3 + 2 = 5$.</p>
<p>Define $g(i)$ to be the smallest positive integer $n$ such that $sf(n) = i$. Though $sf(342)$ is $5$, $sf(25)$ is also... | 8184523820510 | Friday, 4th September 2009, 05:00 pm | 1088 | 75% | hard |
746 | A Messy Dinner | $n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large circular table with $4n$ seats such that men and women alternate.
Let $M(n)$ be the number of ways the families can be seated such that none of the families were seated together.... | $n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large circular table with $4n$ seats such that men and women alternate.
Let $M(n)$ be the number of ways the families can be seated such that none of the families were seated together.... | <p>$n$ families, each with four members, a father, a mother, a son and a daughter, were invited to a restaurant. They were all seated at a large circular table with $4n$ seats such that men and women alternate.</p>
<p>Let $M(n)$ be the number of ways the families can be seated such that none of the families were seated... | 867150922 | Sunday, 7th February 2021, 07:00 am | 309 | 40% | medium |
729 | Range of Periodic Sequence | Consider the sequence of real numbers $a_n$ defined by the starting value $a_0$ and the recurrence
$\displaystyle a_{n+1}=a_n-\frac 1 {a_n}$ for any $n \ge 0$.
For some starting values $a_0$ the sequence will be periodic. For example, $a_0=\sqrt{\frac 1 2}$ yields the sequence:
$\sqrt{\frac 1 2},-\sqrt{\frac 1 2},\sq... | Consider the sequence of real numbers $a_n$ defined by the starting value $a_0$ and the recurrence
$\displaystyle a_{n+1}=a_n-\frac 1 {a_n}$ for any $n \ge 0$.
For some starting values $a_0$ the sequence will be periodic. For example, $a_0=\sqrt{\frac 1 2}$ yields the sequence:
$\sqrt{\frac 1 2},-\sqrt{\frac 1 2},\sq... | <p>Consider the sequence of real numbers $a_n$ defined by the starting value $a_0$ and the recurrence
$\displaystyle a_{n+1}=a_n-\frac 1 {a_n}$ for any $n \ge 0$.</p>
<p>
For some starting values $a_0$ the sequence will be periodic. For example, $a_0=\sqrt{\frac 1 2}$ yields the sequence:
$\sqrt{\frac 1 2},-\sqrt{\fra... | 308896374.2502 | Sunday, 11th October 2020, 05:00 am | 242 | 65% | hard |
353 | Risky Moon | A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$.
There are stations on the moon at the points on the surface of $C(r)$ with integer coordinates. The station at $(0,0,r)$ is called North Pole station, the station at $(0,0,-r)$ is called South Pole station.
All stations are connec... | A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$.
There are stations on the moon at the points on the surface of $C(r)$ with integer coordinates. The station at $(0,0,r)$ is called North Pole station, the station at $(0,0,-r)$ is called South Pole station.
All stations are connec... | <p>
A moon could be described by the sphere $C(r)$ with centre $(0,0,0)$ and radius $r$.
</p>
<p>
There are stations on the moon at the points on the surface of $C(r)$ with integer coordinates. The station at $(0,0,r)$ is called North Pole station, the station at $(0,0,-r)$ is called South Pole station.
</p>
<p>
All s... | 1.2759860331 | Sunday, 9th October 2011, 04:00 am | 554 | 50% | medium |
562 | Maximal Perimeter | Construct triangle $ABC$ such that:
Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin;
the triangle contains no other lattice point inside or on its edges;
the perimeter is maximum.
Let $R$ be the circumradius of triangle $ABC$ and $T(r) = R/r$.
For $r = 5$, one p... | Construct triangle $ABC$ such that:
Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin;
the triangle contains no other lattice point inside or on its edges;
the perimeter is maximum.
Let $R$ be the circumradius of triangle $ABC$ and $T(r) = R/r$.
For $r = 5$, one p... | <p>Construct triangle $ABC$ such that:</p>
<ul><li>Vertices $A$, $B$ and $C$ are lattice points inside or on the circle of radius $r$ centered at the origin;</li>
<li>the triangle contains no other lattice point inside or on its edges;</li>
<li>the perimeter is maximum.</li></ul>
<p>Let $R$ be the circumradius of trian... | 51208732914368 | Sunday, 29th May 2016, 01:00 am | 209 | 75% | hard |
232 | The Race | Two players share an unbiased coin and take it in turns to play The Race.
On Player 1's turn, the coin is tossed once. If it comes up Heads, then Player 1 scores one point; if it comes up Tails, then no points are scored.
On Player 2's turn, a positive integer, $T$, is chosen by Player 2 and the coin is tossed $T$ time... | Two players share an unbiased coin and take it in turns to play The Race.
On Player 1's turn, the coin is tossed once. If it comes up Heads, then Player 1 scores one point; if it comes up Tails, then no points are scored.
On Player 2's turn, a positive integer, $T$, is chosen by Player 2 and the coin is tossed $T$ time... | <p>Two players share an unbiased coin and take it in turns to play <dfn>The Race</dfn>.</p>
<p>On Player 1's turn, the coin is tossed once. If it comes up Heads, then Player 1 scores one point; if it comes up Tails, then no points are scored.</p>
<p>On Player 2's turn, a positive integer, $T$, is chosen by Player 2 and... | 0.83648556 | Friday, 13th February 2009, 05:00 pm | 1991 | 65% | hard |
435 | Polynomials of Fibonacci Numbers | The Fibonacci numbers $\{f_n, n \ge 0\}$ are defined recursively as $f_n = f_{n-1} + f_{n-2}$ with base cases $f_0 = 0$ and $f_1 = 1$.
Define the polynomials $\{F_n, n \ge 0\}$ as $F_n(x) = \displaystyle{\sum_{i=0}^n f_i x^i}$.
For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\,357... | The Fibonacci numbers $\{f_n, n \ge 0\}$ are defined recursively as $f_n = f_{n-1} + f_{n-2}$ with base cases $f_0 = 0$ and $f_1 = 1$.
Define the polynomials $\{F_n, n \ge 0\}$ as $F_n(x) = \displaystyle{\sum_{i=0}^n f_i x^i}$.
For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + 13x^7$, and $F_7(11) = 268\,357... | <p>The <strong>Fibonacci numbers</strong> $\{f_n, n \ge 0\}$ are defined recursively as $f_n = f_{n-1} + f_{n-2}$ with base cases $f_0 = 0$ and $f_1 = 1$.</p>
<p>Define the polynomials $\{F_n, n \ge 0\}$ as $F_n(x) = \displaystyle{\sum_{i=0}^n f_i x^i}$.</p>
<p>For example, $F_7(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6... | 252541322550 | Saturday, 7th September 2013, 04:00 pm | 1266 | 30% | easy |
633 | Square Prime Factors II | For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. It can be shown th... | For an integer $n$, we define the square prime factors of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.
Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factors. It can be shown th... | <p>For an integer $n$, we define the <dfn>square prime factors</dfn> of $n$ to be the primes whose square divides $n$. For example, the square prime factors of $1500=2^2 \times 3 \times 5^3$ are $2$ and $5$.</p>
<p>Let $C_k(N)$ be the number of integers between $1$ and $N$ inclusive with exactly $k$ square prime factor... | 1.0012e-10 | Saturday, 28th July 2018, 01:00 pm | 356 | 50% | medium |
782 | Distinct Rows and Columns | The complexity of an $n\times n$ binary matrix is the number of distinct rows and columns.
For example, consider the $3\times 3$ matrices
$$ \mathbf{A} = \begin{pmatrix} 1&0&1\\0&0&0\\1&0&1\end{pmatrix} \quad
\mathbf{B} = \begin{pmatrix} 0&0&0\\0&0&0\\1&1&1\end{pmatrix} $$
$\mathbf{A}$ has complexity $2$ because th... | The complexity of an $n\times n$ binary matrix is the number of distinct rows and columns.
For example, consider the $3\times 3$ matrices
$$ \mathbf{A} = \begin{pmatrix} 1&0&1\\0&0&0\\1&0&1\end{pmatrix} \quad
\mathbf{B} = \begin{pmatrix} 0&0&0\\0&0&0\\1&1&1\end{pmatrix} $$
$\mathbf{A}$ has complexity $2$ because th... | <p>The <dfn>complexity</dfn> of an $n\times n$ binary matrix is the number of distinct rows and columns.</p>
<p>
For example, consider the $3\times 3$ matrices
$$ \mathbf{A} = \begin{pmatrix} 1&0&1\\0&0&0\\1&0&1\end{pmatrix} \quad
\mathbf{B} = \begin{pmatrix} 0&0&0\\0&0&0\\1&a... | 318313204 | Saturday, 22nd January 2022, 10:00 pm | 164 | 70% | hard |
677 | Coloured Graphs | Let $g(n)$ be the number of undirected graphs with $n$ nodes satisfying the following properties:
The graph is connected and has no cycles or multiple edges.
Each node is either red, blue, or yellow.
A red node may have no more than 4 edges connected to it.
A blue or yellow node may have no more than 3 edges connected... | Let $g(n)$ be the number of undirected graphs with $n$ nodes satisfying the following properties:
The graph is connected and has no cycles or multiple edges.
Each node is either red, blue, or yellow.
A red node may have no more than 4 edges connected to it.
A blue or yellow node may have no more than 3 edges connected... | <p>Let $g(n)$ be the number of <strong>undirected graphs</strong> with $n$ nodes satisfying the following properties:</p>
<ul>
<li>The graph is connected and has no cycles or multiple edges.</li>
<li>Each node is either red, blue, or yellow.</li>
<li>A red node may have no more than 4 edges connected to it.</li>
<li>A ... | 984183023 | Saturday, 29th June 2019, 07:00 pm | 202 | 90% | hard |
551 | Sum of Digits Sequence | Let $a_0, a_1, \dots$ be an integer sequence defined by:
$a_0 = 1$;
for $n \ge 1$, $a_n$ is the sum of the digits of all preceding terms.
The sequence starts with $1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots$
You are given $a_{10^6} = 31054319$.
Find $a_{10^{15}}$. | Let $a_0, a_1, \dots$ be an integer sequence defined by:
$a_0 = 1$;
for $n \ge 1$, $a_n$ is the sum of the digits of all preceding terms.
The sequence starts with $1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots$
You are given $a_{10^6} = 31054319$.
Find $a_{10^{15}}$. | <p>Let $a_0, a_1, \dots$ be an integer sequence defined by:</p>
<ul>
<li>$a_0 = 1$;</li>
<li>for $n \ge 1$, $a_n$ is the sum of the digits of all preceding terms.</li>
</ul>
<p>The sequence starts with $1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots$<br/>
You are given $a_{10^6} = 31054319$.</p>
<p>Find $a_{10^{15}}$.</p> | 73597483551591773 | Saturday, 12th March 2016, 04:00 pm | 518 | 50% | medium |
694 | Cube-full Divisors | A positive integer $n$ is considered cube-full, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cube-full.
Let $s(n)$ be the function that counts the number of cube-full divisors of $n$. For example, $1$, $8$ and $16$ are the three cube-full divisors of $16$. Therefore, $s(16)=3$.
... | A positive integer $n$ is considered cube-full, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cube-full.
Let $s(n)$ be the function that counts the number of cube-full divisors of $n$. For example, $1$, $8$ and $16$ are the three cube-full divisors of $16$. Therefore, $s(16)=3$.
... | <p>
A positive integer $n$ is considered <dfn>cube-full</dfn>, if for every prime $p$ that divides $n$, so does $p^3$. Note that $1$ is considered cube-full.
</p>
<p>
Let $s(n)$ be the function that counts the number of cube-full divisors of $n$. For example, $1$, $8$ and $16$ are the three cube-full divisors of $16$. ... | 1339784153569958487 | Saturday, 21st December 2019, 07:00 pm | 1134 | 15% | easy |
157 | Base-10 Diophantine Reciprocal | Consider the diophantine equation $\frac 1 a + \frac 1 b = \frac p {10^n}$ with $a, b, p, n$ positive integers and $a \le b$.
For $n=1$ this equation has $20$ solutions that are listed below:
\begin{matrix}
\frac 1 1 + \frac 1 1 = \frac{20}{10} & \frac 1 1 + \frac 1 2 = \frac{15}{10} & \frac 1 1 + \frac 1 5 = \frac{12}... | Consider the diophantine equation $\frac 1 a + \frac 1 b = \frac p {10^n}$ with $a, b, p, n$ positive integers and $a \le b$.
For $n=1$ this equation has $20$ solutions that are listed below:
\begin{matrix}
\frac 1 1 + \frac 1 1 = \frac{20}{10} & \frac 1 1 + \frac 1 2 = \frac{15}{10} & \frac 1 1 + \frac 1 5 = \frac{12}... | <p>Consider the diophantine equation $\frac 1 a + \frac 1 b = \frac p {10^n}$ with $a, b, p, n$ positive integers and $a \le b$.<br/>
For $n=1$ this equation has $20$ solutions that are listed below:
\begin{matrix}
\frac 1 1 + \frac 1 1 = \frac{20}{10} & \frac 1 1 + \frac 1 2 = \frac{15}{10} & \frac 1 1 + \frac... | 53490 | Friday, 1st June 2007, 06:00 pm | 3067 | 65% | hard |
29 | Distinct Powers | Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:
\begin{matrix}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{matrix}
If they are then placed in numerical order, with any repeats ... | Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:
\begin{matrix}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{matrix}
If they are then placed in numerical order, with any repeats ... | <p>Consider all integer combinations of $a^b$ for $2 \le a \le 5$ and $2 \le b \le 5$:</p>
\begin{matrix}
2^2=4, &2^3=8, &2^4=16, &2^5=32\\
3^2=9, &3^3=27, &3^4=81, &3^5=243\\
4^2=16, &4^3=64, &4^4=256, &4^5=1024\\
5^2=25, &5^3=125, &5^4=625, &5^5=3125
\end{matrix}
<p>If they are then placed in numerical order, with an... | 9183 | Friday, 25th October 2002, 06:00 pm | 114713 | 5% | easy |
925 | Larger Digit Permutation III | Let $B(n)$ be the smallest number larger than $n$ that can be formed by rearranging digits of $n$, or $0$ if no such number exists. For example, $B(245) = 254$ and $B(542) = 0$.
Define $\displaystyle T(N) = \sum_{n=1}^N B(n^2)$. You are given $T(10)=270$ and $T(100)=335316$.
Find $T(10^{16})$. Give your answer modulo $... | Let $B(n)$ be the smallest number larger than $n$ that can be formed by rearranging digits of $n$, or $0$ if no such number exists. For example, $B(245) = 254$ and $B(542) = 0$.
Define $\displaystyle T(N) = \sum_{n=1}^N B(n^2)$. You are given $T(10)=270$ and $T(100)=335316$.
Find $T(10^{16})$. Give your answer modulo $... | <p>Let $B(n)$ be the smallest number larger than $n$ that can be formed by rearranging digits of $n$, or $0$ if no such number exists. For example, $B(245) = 254$ and $B(542) = 0$.</p>
<p>Define $\displaystyle T(N) = \sum_{n=1}^N B(n^2)$. You are given $T(10)=270$ and $T(100)=335316$.</p>
<p>Find $T(10^{16})$. Give you... | 400034379 | Saturday, 28th December 2024, 07:00 pm | 153 | 55% | medium |
821 | 123-Separable | A set, $S$, of integers is called 123-separable if $S$, $2S$ and $3S$ are disjoint. Here $2S$ and $3S$ are obtained by multiplying all the elements in $S$ by $2$ and $3$ respectively.
Define $F(n)$ to be the maximum number of elements of
$$(S\cup 2S \cup 3S)\cap \{1,2,3,\ldots,n\}$$
where $S$ ranges over all 123-separ... | A set, $S$, of integers is called 123-separable if $S$, $2S$ and $3S$ are disjoint. Here $2S$ and $3S$ are obtained by multiplying all the elements in $S$ by $2$ and $3$ respectively.
Define $F(n)$ to be the maximum number of elements of
$$(S\cup 2S \cup 3S)\cap \{1,2,3,\ldots,n\}$$
where $S$ ranges over all 123-separ... | <p>
A set, $S$, of integers is called <dfn>123-separable</dfn> if $S$, $2S$ and $3S$ are disjoint. Here $2S$ and $3S$ are obtained by multiplying all the elements in $S$ by $2$ and $3$ respectively.</p>
<p>
Define $F(n)$ to be the maximum number of elements of
$$(S\cup 2S \cup 3S)\cap \{1,2,3,\ldots,n\}$$
where $S$ ran... | 9219661511328178 | Saturday, 17th December 2022, 07:00 pm | 161 | 65% | hard |
883 | Remarkable Triangles | In this problem we consider triangles drawn on a hexagonal lattice, where each lattice point in the plane has six neighbouring points equally spaced around it, all distance $1$ away.
We call a triangle remarkable if
All three vertices and its incentre lie on lattice points
At least one of its angles is $60^\circ$
... | In this problem we consider triangles drawn on a hexagonal lattice, where each lattice point in the plane has six neighbouring points equally spaced around it, all distance $1$ away.
We call a triangle remarkable if
All three vertices and its incentre lie on lattice points
At least one of its angles is $60^\circ$
... | <p>
In this problem we consider triangles drawn on a <b>hexagonal lattice</b>, where each lattice point in the plane has six neighbouring points equally spaced around it, all distance $1$ away.</p>
<p>
We call a triangle <i>remarkable</i> if</p>
<ul>
<li>All three vertices and its <b>incentre</b> lie on lattice points<... | 14854003484704 | Sunday, 24th March 2024, 04:00 am | 109 | 95% | hard |
873 | Words with Gaps | Let $W(p,q,r)$ be the number of words that can be formed using the letter A $p$ times, the letter B $q$ times and the letter C $r$ times with the condition that every A is separated from every B by at least two Cs. For example, CACACCBB is a valid word for $W(2,2,4)$ but ACBCACBC is not.
You are given $W(2,2,4)=32$ an... | Let $W(p,q,r)$ be the number of words that can be formed using the letter A $p$ times, the letter B $q$ times and the letter C $r$ times with the condition that every A is separated from every B by at least two Cs. For example, CACACCBB is a valid word for $W(2,2,4)$ but ACBCACBC is not.
You are given $W(2,2,4)=32$ an... | <p>
Let $W(p,q,r)$ be the number of words that can be formed using the letter A $p$ times, the letter B $q$ times and the letter C $r$ times with the condition that every A is separated from every B by at least two Cs. For example, CACACCBB is a valid word for $W(2,2,4)$ but ACBCACBC is not.</p>
<p>
You are given $W(2,... | 735131856 | Sunday, 21st January 2024, 01:00 am | 328 | 25% | easy |
515 | Dissonant Numbers | Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$.
Let $d(p, n, k) = \sum_{i = 1}^n d(p, i, k - 1)$ for $k \ge 1$.
Let $D(a, b, k) = \sum (d(p, p-1, k) \bmod p)$ for all primes $a \le p \lt a + b$.
You are given:
$D(101,1,10) = 45$
$D(10^3,10^2,10^2) = ... | Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$.
Let $d(p, n, k) = \sum_{i = 1}^n d(p, i, k - 1)$ for $k \ge 1$.
Let $D(a, b, k) = \sum (d(p, p-1, k) \bmod p)$ for all primes $a \le p \lt a + b$.
You are given:
$D(101,1,10) = 45$
$D(10^3,10^2,10^2) = ... | <p>Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$.<br/>
Let $d(p, n, k) = \sum_{i = 1}^n d(p, i, k - 1)$ for $k \ge 1$.<br/>
Let $D(a, b, k) = \sum (d(p, p-1, k) \bmod p)$ for all primes $a \le p \lt a + b$.</p>
<p>You are given:</p>
<ul><li>$D(101,1... | 2422639000800 | Sunday, 10th May 2015, 07:00 am | 507 | 40% | medium |
407 | Idempotents | If we calculate $a^2 \bmod 6$ for $0 \leq a \leq 5$ we get: $0,1,4,3,4,1$.
The largest value of $a$ such that $a^2 \equiv a \bmod 6$ is $4$.
Let's call $M(n)$ the largest value of $a \lt n$ such that $a^2 \equiv a \pmod n$.
So $M(6) = 4$.
Find $\sum M(n)$ for $1 \leq n \leq 10^7$. | If we calculate $a^2 \bmod 6$ for $0 \leq a \leq 5$ we get: $0,1,4,3,4,1$.
The largest value of $a$ such that $a^2 \equiv a \bmod 6$ is $4$.
Let's call $M(n)$ the largest value of $a \lt n$ such that $a^2 \equiv a \pmod n$.
So $M(6) = 4$.
Find $\sum M(n)$ for $1 \leq n \leq 10^7$. | <p>
If we calculate $a^2 \bmod 6$ for $0 \leq a \leq 5$ we get: $0,1,4,3,4,1$.
</p>
<p>
The largest value of $a$ such that $a^2 \equiv a \bmod 6$ is $4$.<br/>
Let's call $M(n)$ the largest value of $a \lt n$ such that $a^2 \equiv a \pmod n$.<br/>
So $M(6) = 4$.
</p>
<p>
Find $\sum M(n)$ for $1 \leq n \leq 10^7$.
</p> | 39782849136421 | Sunday, 23rd December 2012, 10:00 am | 2751 | 20% | easy |
718 | Unreachable Numbers | Consider the equation
$17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e.
$a,b,c,p \gt 0$.
For a given $p$ there are some values of $n > 0$ for which the equation cannot be solved. We call these unreachable values.
Define $G(p)$ to be the sum of all unreachable values of $n$ for the given va... | Consider the equation
$17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e.
$a,b,c,p \gt 0$.
For a given $p$ there are some values of $n > 0$ for which the equation cannot be solved. We call these unreachable values.
Define $G(p)$ to be the sum of all unreachable values of $n$ for the given va... | <p>Consider the equation
$17^pa+19^pb+23^pc = n$ where $a$, $b$, $c$ and $p$ are positive integers, i.e.
$a,b,c,p \gt 0$.</p>
<p>For a given $p$ there are some values of $n > 0$ for which the equation cannot be solved. We call these <dfn>unreachable values</dfn>.</p>
<p>Define $G(p)$ to be the sum of all unreachable... | 228579116 | Saturday, 30th May 2020, 05:00 pm | 362 | 35% | medium |
239 | Twenty-two Foolish Primes | A set of disks numbered $1$ through $100$ are placed in a line in random order.
What is the probability that we have a partial derangement such that exactly $22$ prime number discs are found away from their natural positions?
(Any number of non-prime disks may also be found in or out of their natural positions.)
Give y... | A set of disks numbered $1$ through $100$ are placed in a line in random order.
What is the probability that we have a partial derangement such that exactly $22$ prime number discs are found away from their natural positions?
(Any number of non-prime disks may also be found in or out of their natural positions.)
Give y... | <p>A set of disks numbered $1$ through $100$ are placed in a line in random order.</p>
<p>What is the probability that we have a partial derangement such that exactly $22$ prime number discs are found away from their natural positions?<br/>
(Any number of non-prime disks may also be found in or out of their natural pos... | 0.001887854841 | Friday, 3rd April 2009, 05:00 pm | 2019 | 65% | hard |
725 | Digit Sum Numbers | A number where one digit is the sum of the other digits is called a digit sum number or DS-number for short. For example, $352$, $3003$ and $32812$ are DS-numbers.
We define $S(n)$ to be the sum of all DS-numbers of $n$ digits or less.
You are given $S(3) = 63270$ and $S(7) = 85499991450$.
Find $S(2020)$. Give yo... | A number where one digit is the sum of the other digits is called a digit sum number or DS-number for short. For example, $352$, $3003$ and $32812$ are DS-numbers.
We define $S(n)$ to be the sum of all DS-numbers of $n$ digits or less.
You are given $S(3) = 63270$ and $S(7) = 85499991450$.
Find $S(2020)$. Give yo... | <p>
A number where one digit is the sum of the <b>other</b> digits is called a <dfn>digit sum number</dfn> or DS-number for short. For example, $352$, $3003$ and $32812$ are DS-numbers.
</p>
<p>
We define $S(n)$ to be the sum of all DS-numbers of $n$ digits or less.
</p>
<p>
You are given $S(3) = 63270$ and $S(7) = 854... | 4598797036650685 | Saturday, 12th September 2020, 05:00 pm | 1230 | 10% | easy |
528 | Constrained Sums | Let $S(n, k, b)$ represent the number of valid solutions to $x_1 + x_2 + \cdots + x_k \le n$, where $0 \le x_m \le b^m$ for all $1 \le m \le k$.
For example, $S(14,3,2) = 135$, $S(200,5,3) = 12949440$, and $S(1000,10,5) \bmod 1\,000\,000\,007 = 624839075$.
Find $(\sum_{10 \le k \le 15} S(10^k, k, k)) \bmod 1\,000\,000\... | Let $S(n, k, b)$ represent the number of valid solutions to $x_1 + x_2 + \cdots + x_k \le n$, where $0 \le x_m \le b^m$ for all $1 \le m \le k$.
For example, $S(14,3,2) = 135$, $S(200,5,3) = 12949440$, and $S(1000,10,5) \bmod 1\,000\,000\,007 = 624839075$.
Find $(\sum_{10 \le k \le 15} S(10^k, k, k)) \bmod 1\,000\,000\... | <p>Let $S(n, k, b)$ represent the number of valid solutions to $x_1 + x_2 + \cdots + x_k \le n$, where $0 \le x_m \le b^m$ for all $1 \le m \le k$.</p>
<p>For example, $S(14,3,2) = 135$, $S(200,5,3) = 12949440$, and $S(1000,10,5) \bmod 1\,000\,000\,007 = 624839075$.</p>
<p>Find $(\sum_{10 \le k \le 15} S(10^k, k, k)) \... | 779027989 | Saturday, 3rd October 2015, 07:00 pm | 327 | 60% | hard |
582 | Nearly Isosceles $120$ Degree Triangles | Let $a, b$ and $c$ be the sides of an integer sided triangle with one angle of $120$ degrees, $a \le b \le c$ and $b-a \le 100$.
Let $T(n)$ be the number of such triangles with $c \le n$.
$T(1000)=235$ and $T(10^8)=1245$.
Find $T(10^{100})$. | Let $a, b$ and $c$ be the sides of an integer sided triangle with one angle of $120$ degrees, $a \le b \le c$ and $b-a \le 100$.
Let $T(n)$ be the number of such triangles with $c \le n$.
$T(1000)=235$ and $T(10^8)=1245$.
Find $T(10^{100})$. | <p>
Let $a, b$ and $c$ be the sides of an integer sided triangle with one angle of $120$ degrees, $a \le b \le c$ and $b-a \le 100$.<br/>
Let $T(n)$ be the number of such triangles with $c \le n$.<br/>
$T(1000)=235$ and $T(10^8)=1245$.<br/>
Find $T(10^{100})$.
</p> | 19903 | Sunday, 18th December 2016, 10:00 am | 363 | 50% | medium |
396 | Weak Goodstein Sequence | For any positive integer $n$, the $n$th weak Goodstein sequence $\{g_1, g_2, g_3, \dots\}$ is defined as:
$g_1 = n$
for $k \gt 1$, $g_k$ is obtained by writing $g_{k-1}$ in base $k$, interpreting it as a base $k + 1$ number, and subtracting $1$.
The sequence terminates when $g_k$ becomes $0$.
For example, the $6$t... | For any positive integer $n$, the $n$th weak Goodstein sequence $\{g_1, g_2, g_3, \dots\}$ is defined as:
$g_1 = n$
for $k \gt 1$, $g_k$ is obtained by writing $g_{k-1}$ in base $k$, interpreting it as a base $k + 1$ number, and subtracting $1$.
The sequence terminates when $g_k$ becomes $0$.
For example, the $6$t... | <p>
For any positive integer $n$, the <strong>$n$th weak Goodstein sequence</strong> $\{g_1, g_2, g_3, \dots\}$ is defined as:
</p><ul><li> $g_1 = n$
</li><li> for $k \gt 1$, $g_k$ is obtained by writing $g_{k-1}$ in base $k$, interpreting it as a base $k + 1$ number, and subtracting $1$.
</li></ul>
The sequence termin... | 173214653 | Sunday, 30th September 2012, 02:00 am | 728 | 40% | medium |
320 | Factorials Divisible by a Huge Integer | Let $N(i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$
Let $S(u)=\sum N(i)$ for $10 \le i \le u$.
$S(1000)=614538266565663$.
Find $S(1\,000\,000) \bmod 10^{18}$. | Let $N(i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$
Let $S(u)=\sum N(i)$ for $10 \le i \le u$.
$S(1000)=614538266565663$.
Find $S(1\,000\,000) \bmod 10^{18}$. | <p>
Let $N(i)$ be the smallest integer $n$ such that $n!$ is divisible by $(i!)^{1234567890}$</p>
<p>
Let $S(u)=\sum N(i)$ for $10 \le i \le u$.
</p>
<p>
$S(1000)=614538266565663$.
</p>
<p>
Find $S(1\,000\,000) \bmod 10^{18}$.
</p> | 278157919195482643 | Saturday, 15th January 2011, 10:00 pm | 987 | 50% | medium |
102 | Triangle Containment | Three distinct points are plotted at random on a Cartesian plane, for which $-1000 \le x, y \le 1000$, such that a triangle is formed.
Consider the following two triangles:
\begin{gather}
A(-340,495), B(-153,-910), C(835,-947)\\
X(-175,41), Y(-421,-714), Z(574,-645)
\end{gather}
It can be verified that triangle $ABC$ c... | Three distinct points are plotted at random on a Cartesian plane, for which $-1000 \le x, y \le 1000$, such that a triangle is formed.
Consider the following two triangles:
\begin{gather}
A(-340,495), B(-153,-910), C(835,-947)\\
X(-175,41), Y(-421,-714), Z(574,-645)
\end{gather}
It can be verified that triangle $ABC$ c... | <p>Three distinct points are plotted at random on a Cartesian plane, for which $-1000 \le x, y \le 1000$, such that a triangle is formed.</p>
<p>Consider the following two triangles:</p>
\begin{gather}
A(-340,495), B(-153,-910), C(835,-947)\\
X(-175,41), Y(-421,-714), Z(574,-645)
\end{gather}
<p>It can be verified that... | 228 | Friday, 12th August 2005, 06:00 pm | 24175 | 15% | easy |
147 | Rectangles in Cross-hatched Grids | In a $3 \times 2$ cross-hatched grid, a total of $37$ different rectangles could be situated within that grid as indicated in the sketch.
There are $5$ grids smaller than $3 \times 2$, vertical and horizontal dimensions being important, i.e. $1 \times 1$, $2 \times 1$, $3 \times 1$, $1 \times 2$ and $2 \times 2$. If e... | In a $3 \times 2$ cross-hatched grid, a total of $37$ different rectangles could be situated within that grid as indicated in the sketch.
There are $5$ grids smaller than $3 \times 2$, vertical and horizontal dimensions being important, i.e. $1 \times 1$, $2 \times 1$, $3 \times 1$, $1 \times 2$ and $2 \times 2$. If e... | <p>In a $3 \times 2$ cross-hatched grid, a total of $37$ different rectangles could be situated within that grid as indicated in the sketch.</p>
<div class="center"><img alt="" class="dark_img" src="resources/images/0147.png?1678992052"/></div>
<p>There are $5$ grids smaller than $3 \times 2$, vertical and horizontal d... | 846910284 | Saturday, 31st March 2007, 06:00 am | 3428 | 65% | hard |
72 | Counting Fractions | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \f... | Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.
If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \f... | <p>Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.</p>
<p>If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:
$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \fr... | 303963552391 | Friday, 18th June 2004, 06:00 pm | 25006 | 20% | easy |
695 | Random Rectangles | Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square and a diagonal that is one of the three line segments $\overline{P_1P_2}$, $\overline{P_1P_3}$ or $\overline{P_2P_3}$ (see picture below).
We are interest... | Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square and a diagonal that is one of the three line segments $\overline{P_1P_2}$, $\overline{P_1P_3}$ or $\overline{P_2P_3}$ (see picture below).
We are interest... | <p>Three points, $P_1$, $P_2$ and $P_3$, are randomly selected within a unit square. Consider the three rectangles with sides parallel to the sides of the unit square and a diagonal that is one of the three line segments $\overline{P_1P_2}$, $\overline{P_1P_3}$ or $\overline{P_2P_3}$ (see picture below).</p>
<div class... | 0.1017786859 | Saturday, 28th December 2019, 10:00 pm | 233 | 70% | hard |
78 | Coin Partitions | Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so $p(5)=7$.
OOOOO
OOOO O
OOO OO
OOO O O
OO OO O
OO O O O
O O O O O
Find the least value of $n$ for which $p(... | Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so $p(5)=7$.
OOOOO
OOOO O
OOO OO
OOO O O
OO OO O
OO O O O
O O O O O
Find the least value of $n$ for which $p(... | <p>Let $p(n)$ represent the number of different ways in which $n$ coins can be separated into piles. For example, five coins can be separated into piles in exactly seven different ways, so $p(5)=7$.</p>
<div class="margin_left">
OOOOO<br/>
OOOO O<br/>
OOO OO<br/>
OOO O O<br/>
OO OO O<br/>
OO O O O<br/... | 55374 | Friday, 10th September 2004, 06:00 pm | 19163 | 30% | easy |
871 | Drifting Subsets | Let $f$ be a function from a finite set $S$ to itself. A drifting subset for $f$ is a subset $A$ of $S$ such that the number of elements in the union $A \cup f(A)$ is equal to twice the number of elements of $A$.
We write $D(f)$ for the maximal number of elements among all drifting subsets for $f$.
For a positive inte... | Let $f$ be a function from a finite set $S$ to itself. A drifting subset for $f$ is a subset $A$ of $S$ such that the number of elements in the union $A \cup f(A)$ is equal to twice the number of elements of $A$.
We write $D(f)$ for the maximal number of elements among all drifting subsets for $f$.
For a positive inte... | <p>
Let $f$ be a function from a finite set $S$ to itself. A <dfn>drifting subset</dfn> for $f$ is a subset $A$ of $S$ such that the number of elements in the union $A \cup f(A)$ is equal to twice the number of elements of $A$.<br/>
We write $D(f)$ for the maximal number of elements among all drifting subsets for $f$.<... | 2848790 | Saturday, 6th January 2024, 07:00 pm | 348 | 25% | easy |
656 | Palindromic Sequences | Given an irrational number $\alpha$, let $S_\alpha(n)$ be the sequence $S_\alpha(n)=\lfloor {\alpha \cdot n} \rfloor - \lfloor {\alpha \cdot (n-1)} \rfloor$ for $n \ge 1$.
($\lfloor \cdots \rfloor$ is the floor-function.)
It can be proven that for any irrational $\alpha$ there exist infinitely many values of $n$ suc... | Given an irrational number $\alpha$, let $S_\alpha(n)$ be the sequence $S_\alpha(n)=\lfloor {\alpha \cdot n} \rfloor - \lfloor {\alpha \cdot (n-1)} \rfloor$ for $n \ge 1$.
($\lfloor \cdots \rfloor$ is the floor-function.)
It can be proven that for any irrational $\alpha$ there exist infinitely many values of $n$ suc... | <p>
Given an irrational number $\alpha$, let $S_\alpha(n)$ be the sequence $S_\alpha(n)=\lfloor {\alpha \cdot n} \rfloor - \lfloor {\alpha \cdot (n-1)} \rfloor$ for $n \ge 1$.<br/>
($\lfloor \cdots \rfloor$ is the floor-function.)
</p>
<p>
It can be proven that for any irrational $\alpha$ there exist infinitely many v... | 888873503555187 | Sunday, 17th February 2019, 10:00 am | 261 | 50% | medium |
137 | Fibonacci Golden Nuggets | Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.
For this problem we shall be interested in values of $x$ for which $A_F(x)$ is a positive... | Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.
For this problem we shall be interested in values of $x$ for which $A_F(x)$ is a positive... | <p>Consider the infinite polynomial series $A_F(x) = x F_1 + x^2 F_2 + x^3 F_3 + \dots$, where $F_k$ is the $k$th term in the Fibonacci sequence: $1, 1, 2, 3, 5, 8, \dots$; that is, $F_k = F_{k-1} + F_{k-2}$, $F_1 = 1$ and $F_2 = 1$.</p>
<p>For this problem we shall be interested in values of $x$ for which $A_F(x)$ is ... | 1120149658760 | Friday, 12th January 2007, 06:00 pm | 6255 | 50% | medium |
797 | Cyclogenic Polynomials | A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to $1$.
Define $\mathcal{F}$ to be the set of all monic polynomials with integer coefficients (including the constant polynomial $p(x)=1$). A polynomial $p(x)\in\mathcal{F}$ is cyclogenic if there exists $q(x)\in\math... | A monic polynomial is a single-variable polynomial in which the coefficient of highest degree is equal to $1$.
Define $\mathcal{F}$ to be the set of all monic polynomials with integer coefficients (including the constant polynomial $p(x)=1$). A polynomial $p(x)\in\mathcal{F}$ is cyclogenic if there exists $q(x)\in\math... | <p>A <strong>monic polynomial</strong> is a single-variable polynomial in which the coefficient of highest degree is equal to $1$.</p>
<p>Define $\mathcal{F}$ to be the set of all monic polynomials with integer coefficients (including the constant polynomial $p(x)=1$). A polynomial $p(x)\in\mathcal{F}$ is <dfn>cyclogen... | 47722272 | Saturday, 7th May 2022, 08:00 pm | 201 | 50% | medium |
369 | Badugi | In a standard $52$ card deck of playing cards, a set of $4$ cards is a Badugi if it contains $4$ cards with no pairs and no two cards of the same suit.
Let $f(n)$ be the number of ways to choose $n$ cards with a $4$ card subset that is a Badugi. For example, there are $2598960$ ways to choose five cards from a standar... | In a standard $52$ card deck of playing cards, a set of $4$ cards is a Badugi if it contains $4$ cards with no pairs and no two cards of the same suit.
Let $f(n)$ be the number of ways to choose $n$ cards with a $4$ card subset that is a Badugi. For example, there are $2598960$ ways to choose five cards from a standar... | <p>In a standard $52$ card deck of playing cards, a set of $4$ cards is a <strong>Badugi</strong> if it contains $4$ cards with no pairs and no two cards of the same suit.</p>
<p>Let $f(n)$ be the number of ways to choose $n$ cards with a $4$ card subset that is a Badugi. For example, there are $2598960$ ways to choos... | 862400558448 | Sunday, 29th January 2012, 04:00 am | 527 | 60% | hard |
1 | Multiples of 3 or 5 | If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3, 5, 6$ and $9$. The sum of these multiples is $23$.
Find the sum of all the multiples of $3$ or $5$ below $1000$. | If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3, 5, 6$ and $9$. The sum of these multiples is $23$.
Find the sum of all the multiples of $3$ or $5$ below $1000$. | <p>If we list all the natural numbers below $10$ that are multiples of $3$ or $5$, we get $3, 5, 6$ and $9$. The sum of these multiples is $23$.</p>
<p>Find the sum of all the multiples of $3$ or $5$ below $1000$.</p> | 233168 | Friday, 5th October 2001, 06:00 pm | 1025612 | 5% | easy |
231 | Prime Factorisation of Binomial Coefficients | The binomial coefficient $\displaystyle \binom {10} 3 = 120$.
$120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and $2 + 2 + 2 + 3 + 5 = 14$.
So the sum of the terms in the prime factorisation of $\displaystyle \binom {10} 3$ is $14$.
Find the sum of the terms in the prime factorisation of $\displ... | The binomial coefficient $\displaystyle \binom {10} 3 = 120$.
$120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and $2 + 2 + 2 + 3 + 5 = 14$.
So the sum of the terms in the prime factorisation of $\displaystyle \binom {10} 3$ is $14$.
Find the sum of the terms in the prime factorisation of $\displ... | <p>The binomial coefficient $\displaystyle \binom {10} 3 = 120$.<br/>
$120 = 2^3 \times 3 \times 5 = 2 \times 2 \times 2 \times 3 \times 5$, and $2 + 2 + 2 + 3 + 5 = 14$.<br/>
So the sum of the terms in the prime factorisation of $\displaystyle \binom {10} 3$ is $14$.
<br/><br/>
Find the sum of the terms in the prime f... | 7526965179680 | Friday, 6th February 2009, 01:00 pm | 5961 | 40% | medium |
282 | The Ackermann Function | $\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}}$
For non-negative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows:
$$
A(m,n) = \cases{
n+1 &$\htmltext{ if }m=0$\cr
A(m-1,1) &$\htmltext{ if }m>0 \htmltext{ and } n=0$\cr
A(m-1,A(m,n-1)) &$\htmltext{ if }m>0 \htmltext{ and } n... | $\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}}$
For non-negative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows:
$$
A(m,n) = \cases{
n+1 &$\htmltext{ if }m=0$\cr
A(m-1,1) &$\htmltext{ if }m>0 \htmltext{ and } n=0$\cr
A(m-1,A(m,n-1)) &$\htmltext{ if }m>0 \htmltext{ and } n... | $\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}}$
<p>
For non-negative integers $m$, $n$, the Ackermann function $A(m,n)$ is defined as follows:
$$
A(m,n) = \cases{
n+1 &$\htmltext{ if }m=0$\cr
A(m-1,1) &$\htmltext{ if }m>0 \htmltext{ and } n=0$\cr
A(m-1,A(m,n-1)) &$\htmltext{ if }m>0... | 1098988351 | Friday, 12th March 2010, 05:00 pm | 1155 | 70% | hard |
628 | Open Chess Positions | A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which $n$ pawns are placed on a $n \times n$
board in such a way, that there is a single pawn in every row and every column.
We call such a position an open posit... | A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which $n$ pawns are placed on a $n \times n$
board in such a way, that there is a single pawn in every row and every column.
We call such a position an open posit... | <p>
A position in chess is an (orientated) arrangement of chess pieces placed on a chessboard of given size. In the following, we consider all positions in which $n$ pawns are placed on a $n \times n$
board in such a way, that there is a single pawn in every row and every column.
</p>
<p>
We call such a position an... | 210286684 | Sunday, 3rd June 2018, 01:00 am | 870 | 30% | easy |
93 | Arithmetic Expressions | By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$) and brackets/parentheses, it is possible to form different positive integer targets.
For example,
\begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \times (3 + 1 / 2)\\
19 &= ... | By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$) and brackets/parentheses, it is possible to form different positive integer targets.
For example,
\begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \times (3 + 1 / 2)\\
19 &= ... | <p>By using each of the digits from the set, $\{1, 2, 3, 4\}$, exactly once, and making use of the four arithmetic operations ($+, -, \times, /$) and brackets/parentheses, it is possible to form different positive integer targets.</p>
<p>For example,</p>
\begin{align}
8 &= (4 \times (1 + 3)) / 2\\
14 &= 4 \times (3 + 1... | 1258 | Friday, 15th April 2005, 06:00 pm | 13412 | 35% | medium |
303 | Multiples with Small Digits | For a positive integer $n$, define $f(n)$ as the least positive multiple of $n$ that, written in base $10$, uses only digits $\le 2$.
Thus $f(2)=2$, $f(3)=12$, $f(7)=21$, $f(42)=210$, $f(89)=1121222$.
Also, $\sum \limits_{n = 1}^{100} {\dfrac{f(n)}{n}} = 11363107$.
Find $\sum \limits_{n=1}^{10000} {\dfrac{f(n)}{n}}$. | For a positive integer $n$, define $f(n)$ as the least positive multiple of $n$ that, written in base $10$, uses only digits $\le 2$.
Thus $f(2)=2$, $f(3)=12$, $f(7)=21$, $f(42)=210$, $f(89)=1121222$.
Also, $\sum \limits_{n = 1}^{100} {\dfrac{f(n)}{n}} = 11363107$.
Find $\sum \limits_{n=1}^{10000} {\dfrac{f(n)}{n}}$. | <p>
For a positive integer $n$, define $f(n)$ as the least positive multiple of $n$ that, written in base $10$, uses only digits $\le 2$.</p>
<p>Thus $f(2)=2$, $f(3)=12$, $f(7)=21$, $f(42)=210$, $f(89)=1121222$.</p>
<p>Also, $\sum \limits_{n = 1}^{100} {\dfrac{f(n)}{n}} = 11363107$.</p>
<p>
Find $\sum \limits_{n=1}^{10... | 1111981904675169 | Saturday, 25th September 2010, 10:00 pm | 3976 | 35% | medium |
364 | Comfortable Distance | There are $N$ seats in a row. $N$ people come after each other to fill the seats according to the following rules:
If there is any seat whose adjacent seat(s) are not occupied take such a seat.
If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.
Otherwise take o... | There are $N$ seats in a row. $N$ people come after each other to fill the seats according to the following rules:
If there is any seat whose adjacent seat(s) are not occupied take such a seat.
If there is no such seat and there is any seat for which only one adjacent seat is occupied take such a seat.
Otherwise take o... | <p>
There are $N$ seats in a row. $N$ people come after each other to fill the seats according to the following rules:
</p><ol type="1"><li>If there is any seat whose adjacent seat(s) are not occupied take such a seat.</li>
<li>If there is no such seat and there is any seat for which only one adjacent seat is occupied ... | 44855254 | Saturday, 24th December 2011, 01:00 pm | 784 | 50% | medium |
659 | Largest Prime | Consider the sequence $n^2+3$ with $n \ge 1$.
If we write down the first terms of this sequence we get:
$4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, 172, 199, 228, 259, 292, 327, 364, \dots$ .
We see that the terms for $n=6$ and $n=7$ ($39$ and $52$) are both divisible by $13$.
In fact $13$ is the largest prime... | Consider the sequence $n^2+3$ with $n \ge 1$.
If we write down the first terms of this sequence we get:
$4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, 172, 199, 228, 259, 292, 327, 364, \dots$ .
We see that the terms for $n=6$ and $n=7$ ($39$ and $52$) are both divisible by $13$.
In fact $13$ is the largest prime... | <p>
Consider the sequence $n^2+3$ with $n \ge 1$. <br/>
If we write down the first terms of this sequence we get:<br/>
$4, 7, 12, 19, 28, 39, 52, 67, 84, 103, 124, 147, 172, 199, 228, 259, 292, 327, 364, \dots$ .<br/>
We see that the terms for $n=6$ and $n=7$ ($39$ and $52$) are both divisible by $13$.<br/>
In fact $... | 238518915714422000 | Saturday, 2nd March 2019, 04:00 pm | 1087 | 20% | easy |
684 | Inverse Digit Sum | Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.
Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i-2}+f_{i-1}$ for all $i \ge 2$.
Find $\displaystyle \sum_{i=2}^{... | Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.
Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i-2}+f_{i-1}$ for all $i \ge 2$.
Find $\displaystyle \sum_{i=2}^{... | <p>Define $s(n)$ to be the smallest number that has a digit sum of $n$. For example $s(10) = 19$.<br>
Let $\displaystyle S(k) = \sum_{n=1}^k s(n)$. You are given $S(20) = 1074$.</br></p>
<p>
Further let $f_i$ be the Fibonacci sequence defined by $f_0=0, f_1=1$ and $f_i=f_{i-2}+f_{i-1}$ for all $i \ge 2$.</p>
<p>
Find $... | 922058210 | Saturday, 19th October 2019, 04:00 pm | 3024 | 5% | easy |
564 | Maximal Polygons | A line segment of length $2n-3$ is randomly split into $n$ segments of integer length ($n \ge 3$). In the sequence given by this split, the segments are then used as consecutive sides of a convex $n$-polygon, formed in such a way that its area is maximal. All of the $\binom{2n-4} {n-1}$ possibilities for splitting up ... | A line segment of length $2n-3$ is randomly split into $n$ segments of integer length ($n \ge 3$). In the sequence given by this split, the segments are then used as consecutive sides of a convex $n$-polygon, formed in such a way that its area is maximal. All of the $\binom{2n-4} {n-1}$ possibilities for splitting up ... | <p>A line segment of length $2n-3$ is randomly split into $n$ segments of integer length ($n \ge 3$). In the sequence given by this split, the segments are then used as consecutive sides of a convex $n$-polygon, formed in such a way that its area is maximal. All of the $\binom{2n-4} {n-1}$ possibilities for splitting ... | 12363.698850 | Sunday, 12th June 2016, 07:00 am | 267 | 60% | hard |
165 | Intersections | A segment is uniquely defined by its two endpoints. By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is... | A segment is uniquely defined by its two endpoints. By considering two line segments in plane geometry there are three possibilities:
the segments have zero points, one point, or infinitely many points in common.
Moreover when two segments have exactly one point in common it might be the case that that common point is... | <p>A segment is uniquely defined by its two endpoints.<br/> By considering two line segments in plane geometry there are three possibilities:<br/>
the segments have zero points, one point, or infinitely many points in common.</p>
<p>Moreover when two segments have exactly one point in common it might be the case that ... | 2868868 | Saturday, 27th October 2007, 10:00 am | 2939 | 65% | hard |
752 | Powers of $1+\sqrt 7$ | When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$.
We write $(1+\sqrt 7)^n = \alpha(n) + \beta(n)\sqrt 7$.
For a given number $x$ we define $g(x)$ to be the smallest positive integer $n$ such that:
$$\begin{align}
\alpha(n) &\equiv 1 \pmod x\qquad \text{and }\\... | When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$.
We write $(1+\sqrt 7)^n = \alpha(n) + \beta(n)\sqrt 7$.
For a given number $x$ we define $g(x)$ to be the smallest positive integer $n$ such that:
$$\begin{align}
\alpha(n) &\equiv 1 \pmod x\qquad \text{and }\\... | <p>
When $(1+\sqrt 7)$ is raised to an integral power, $n$, we always get a number of the form $(a+b\sqrt 7)$.<br/>
We write $(1+\sqrt 7)^n = \alpha(n) + \beta(n)\sqrt 7$.
</p>
<p>
For a given number $x$ we define $g(x)$ to be the smallest positive integer $n$ such that:
$$\begin{align}
\alpha(n) &\equiv 1 \pmod x... | 5610899769745488 | Sunday, 21st March 2021, 01:00 am | 725 | 25% | easy |
249 | Prime Subset Sums | Let $S = \{2, 3, 5, \dots, 4999\}$ be the set of prime numbers less than $5000$.
Find the number of subsets of $S$, the sum of whose elements is a prime number.
Enter the rightmost $16$ digits as your answer. | Let $S = \{2, 3, 5, \dots, 4999\}$ be the set of prime numbers less than $5000$.
Find the number of subsets of $S$, the sum of whose elements is a prime number.
Enter the rightmost $16$ digits as your answer. | <p>Let $S = \{2, 3, 5, \dots, 4999\}$ be the set of prime numbers less than $5000$.</p>
<p>Find the number of subsets of $S$, the sum of whose elements is a prime number.<br/>
Enter the rightmost $16$ digits as your answer.</p> | 9275262564250418 | Saturday, 13th June 2009, 05:00 am | 2810 | 60% | hard |
127 | abc-hits | The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:
$\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1... | The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.
We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:
$\gcd(a, b) = \gcd(a, c) = \gcd(b, c) = 1... | <p>The radical of $n$, $\operatorname{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) = 2 \times 3 \times 7 = 42$.</p>
<p>We shall define the triplet of positive integers $(a, b, c)$ to be an abc-hit if:</p>
<ol><li>$\gcd(a, b) = \gcd(... | 18407904 | Friday, 1st September 2006, 06:00 pm | 7059 | 50% | medium |
841 | Regular Star Polygons | The regular star polygon $\{p/q\}$, for coprime integers $p,q$ with $p \gt 2q \gt 0$, is a polygon formed from $p$ edges of equal length and equal internal angles, such that tracing the complete polygon wraps $q$ times around the centre. For example, $\{8/3\}$ is illustrated below:
The edges of a regular star polygon ... | The regular star polygon $\{p/q\}$, for coprime integers $p,q$ with $p \gt 2q \gt 0$, is a polygon formed from $p$ edges of equal length and equal internal angles, such that tracing the complete polygon wraps $q$ times around the centre. For example, $\{8/3\}$ is illustrated below:
The edges of a regular star polygon ... | <p>The regular star polygon $\{p/q\}$, for coprime integers $p,q$ with $p \gt 2q \gt 0$, is a polygon formed from $p$ edges of equal length and equal internal angles, such that tracing the complete polygon wraps $q$ times around the centre. For example, $\{8/3\}$ is illustrated below:</p>
<div align="center"><img alt="... | 381.7860132854 | Sunday, 30th April 2023, 05:00 am | 203 | 45% | medium |
329 | Prime Frog | Susan has a prime frog.
Her frog is jumping around over $500$ squares numbered $1$ to $500$.
He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range $[1;500]$.(if it lands at either end, it automatically jumps to the only available square on the next move.)
... | Susan has a prime frog.
Her frog is jumping around over $500$ squares numbered $1$ to $500$.
He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range $[1;500]$.(if it lands at either end, it automatically jumps to the only available square on the next move.)
... | <p>Susan has a prime frog.<br/>
Her frog is jumping around over $500$ squares numbered $1$ to $500$.
He can only jump one square to the left or to the right, with equal probability, and he cannot jump outside the range $[1;500]$.<br/>(if it lands at either end, it automatically jumps to the only available square on the... | 199740353/29386561536000 | Sunday, 20th March 2011, 01:00 am | 2762 | 25% | easy |
351 | Hexagonal Orchards | A hexagonal orchard of order $n$ is a triangular lattice made up of points within a regular hexagon with side $n$. The following is an example of a hexagonal orchard of order $5$:
Highlighted in green are the points which are hidden from the center by a point closer to it. It can be seen that for a hexagonal orchar... | A hexagonal orchard of order $n$ is a triangular lattice made up of points within a regular hexagon with side $n$. The following is an example of a hexagonal orchard of order $5$:
Highlighted in green are the points which are hidden from the center by a point closer to it. It can be seen that for a hexagonal orchar... | <p>A <dfn>hexagonal orchard</dfn> of order $n$ is a triangular lattice made up of points within a regular hexagon with side $n$. The following is an example of a hexagonal orchard of order $5$:
</p>
<div align="center">
<img alt="0351_hexorchard.png" class="dark_img" src="resources/images/0351_hexorchard.png?1678992052... | 11762187201804552 | Saturday, 17th September 2011, 10:00 pm | 2850 | 25% | easy |
806 | Nim on Towers of Hanoi | This problem combines the game of Nim with the Towers of Hanoi. For a brief introduction to the rules of these games, please refer to Problem 301 and Problem 497, respectively.
The unique shortest solution to the Towers of Hanoi problem with $n$ disks and $3$ pegs requires $2^n-1$ moves. Number the positions in the sol... | This problem combines the game of Nim with the Towers of Hanoi. For a brief introduction to the rules of these games, please refer to Problem 301 and Problem 497, respectively.
The unique shortest solution to the Towers of Hanoi problem with $n$ disks and $3$ pegs requires $2^n-1$ moves. Number the positions in the sol... | <p>This problem combines the game of Nim with the Towers of Hanoi. For a brief introduction to the rules of these games, please refer to <a href="problem=301">Problem 301</a> and <a href="problem=497">Problem 497</a>, respectively.</p>
<p>The unique shortest solution to the Towers of Hanoi problem with $n$ disks and $3... | 94394343 | Saturday, 9th July 2022, 11:00 pm | 143 | 100% | hard |
706 | $3$-Like Numbers | For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string "2573" has $10$ non-empty substrings, three of which represent numbers that are divisible by $3$, namely $57$, $573$ and $3$. So $f(2573) = 3$.
If $f(n)$ is divisible by $3$ the... | For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string "2573" has $10$ non-empty substrings, three of which represent numbers that are divisible by $3$, namely $57$, $573$ and $3$. So $f(2573) = 3$.
If $f(n)$ is divisible by $3$ the... | <p>
For a positive integer $n$, define $f(n)$ to be the number of non-empty substrings of $n$ that are divisible by $3$. For example, the string "2573" has $10$ non-empty substrings, three of which represent numbers that are divisible by $3$, namely $57$, $573$ and $3$. So $f(2573) = 3$.
</p>
<p>
If $f(n)$ is divisible... | 884837055 | Sunday, 15th March 2020, 07:00 am | 634 | 25% | easy |
499 | St. Petersburg Lottery | A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games.
Each game costs $m$ pounds to play and starts with an initial pot of $1$ pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. When a tail... | A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games.
Each game costs $m$ pounds to play and starts with an initial pot of $1$ pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. When a tail... | <p>A gambler decides to participate in a special lottery. In this lottery the gambler plays a series of one or more games.<br/>
Each game costs $m$ pounds to play and starts with an initial pot of $1$ pound. The gambler flips an unbiased coin. Every time a head appears, the pot is doubled and the gambler continues. Whe... | 0.8660312 | Sunday, 25th January 2015, 10:00 am | 385 | 100% | hard |
307 | Chip Defects | $k$ defects are randomly distributed amongst $n$ integrated-circuit chips produced by a factory (any number of defects may be found on a chip and each defect is independent of the other defects).
Let $p(k, n)$ represent the probability that there is a chip with at least $3$ defects.
For instance $p(3,7) \approx 0.020... | $k$ defects are randomly distributed amongst $n$ integrated-circuit chips produced by a factory (any number of defects may be found on a chip and each defect is independent of the other defects).
Let $p(k, n)$ represent the probability that there is a chip with at least $3$ defects.
For instance $p(3,7) \approx 0.020... | <p>
$k$ defects are randomly distributed amongst $n$ integrated-circuit chips produced by a factory (any number of defects may be found on a chip and each defect is independent of the other defects).
</p>
<p>
Let $p(k, n)$ represent the probability that there is a chip with at least $3$ defects.<br/>
For instance $p(3,... | 0.7311720251 | Sunday, 24th October 2010, 10:00 am | 1840 | 40% | medium |
421 | Prime Factors of $n^{15}+1$ | Numbers of the form $n^{15}+1$ are composite for every integer $n \gt 1$.
For positive integers $n$ and $m$ let $s(n,m)$ be defined as the sum of the distinct prime factors of $n^{15}+1$ not exceeding $m$.
E.g. $2^{15}+1 = 3 \times 3 \times 11 \times 331$.
So $s(2,10) = 3$ and $s(2,1000) = 3+11+331 = 345$.
Also $10^{... | Numbers of the form $n^{15}+1$ are composite for every integer $n \gt 1$.
For positive integers $n$ and $m$ let $s(n,m)$ be defined as the sum of the distinct prime factors of $n^{15}+1$ not exceeding $m$.
E.g. $2^{15}+1 = 3 \times 3 \times 11 \times 331$.
So $s(2,10) = 3$ and $s(2,1000) = 3+11+331 = 345$.
Also $10^{... | <p>
Numbers of the form $n^{15}+1$ are composite for every integer $n \gt 1$.<br/>
For positive integers $n$ and $m$ let $s(n,m)$ be defined as the sum of the <i>distinct</i> prime factors of $n^{15}+1$ not exceeding $m$.
</p>
E.g. $2^{15}+1 = 3 \times 3 \times 11 \times 331$.<br/>
So $s(2,10) = 3$ and $s(2,1000) = 3+1... | 2304215802083466198 | Sunday, 31st March 2013, 04:00 am | 745 | 50% | medium |
803 | Pseudorandom Sequence | Rand48 is a pseudorandom number generator used by some programming languages. It generates a sequence from any given integer $0 \le a_0 < 2^{48}$ using the rule $a_n = (25214903917 \cdot a_{n - 1} + 11) \bmod 2^{48}$.
Let $b_n = \lfloor a_n / 2^{16} \rfloor \bmod 52$.
The sequence $b_0, b_1, \dots$ is translated to a... | Rand48 is a pseudorandom number generator used by some programming languages. It generates a sequence from any given integer $0 \le a_0 < 2^{48}$ using the rule $a_n = (25214903917 \cdot a_{n - 1} + 11) \bmod 2^{48}$.
Let $b_n = \lfloor a_n / 2^{16} \rfloor \bmod 52$.
The sequence $b_0, b_1, \dots$ is translated to a... | <p>
<b>Rand48</b> is a pseudorandom number generator used by some programming languages. It generates a sequence from any given integer $0 \le a_0 < 2^{48}$ using the rule $a_n = (25214903917 \cdot a_{n - 1} + 11) \bmod 2^{48}$.
</p>
<p>
Let $b_n = \lfloor a_n / 2^{16} \rfloor \bmod 52$.
The sequence $b_0, b_1, \dot... | 9300900470636 | Saturday, 18th June 2022, 02:00 pm | 239 | 55% | medium |
920 | Tau Numbers | For a positive integer $n$ we define $\tau(n)$ to be the count of the divisors of $n$. For example, the divisors of $12$ are $\{1,2,3,4,6,12\}$ and so $\tau(12) = 6$.
A positive integer $n$ is a tau number if it is divisible by $\tau(n)$. For example $\tau(12)=6$ and $6$ divides $12$ so $12$ is a tau number.
Let $m(k... | For a positive integer $n$ we define $\tau(n)$ to be the count of the divisors of $n$. For example, the divisors of $12$ are $\{1,2,3,4,6,12\}$ and so $\tau(12) = 6$.
A positive integer $n$ is a tau number if it is divisible by $\tau(n)$. For example $\tau(12)=6$ and $6$ divides $12$ so $12$ is a tau number.
Let $m(k... | <p>For a positive integer $n$ we define $\tau(n)$ to be the count of the divisors of $n$. For example, the divisors of $12$ are $\{1,2,3,4,6,12\}$ and so $\tau(12) = 6$.</p>
<p>
A positive integer $n$ is a <b>tau number</b> if it is divisible by $\tau(n)$. For example $\tau(12)=6$ and $6$ divides $12$ so $12$ is a tau ... | 1154027691000533893 | Sunday, 8th December 2024, 10:00 am | 245 | 30% | easy |
186 | Connectedness of a Network | Here are the records from a busy telephone system with one million users:
RecNrCallerCalled
$1$$200007$$100053$$2$$600183$$500439$$3$$600863$$701497$$\cdots$$\cdots$$\cdots$
The telephone number of the caller and the called number in record $n$ are $\operatorname{Caller}(n) = S_{2n-1}$ and $\operatorname{Called}(n) = ... | Here are the records from a busy telephone system with one million users:
RecNrCallerCalled
$1$$200007$$100053$$2$$600183$$500439$$3$$600863$$701497$$\cdots$$\cdots$$\cdots$
The telephone number of the caller and the called number in record $n$ are $\operatorname{Caller}(n) = S_{2n-1}$ and $\operatorname{Called}(n) = ... | <p>Here are the records from a busy telephone system with one million users:</p>
<div class="center">
<table class="grid" style="margin:0 auto;"><tr><th>RecNr</th><th align="center" width="60">Caller</th><th align="center" width="60">Called</th></tr>
<tr><td align="center">$1$</td><td align="center">$200007$</td><td al... | 2325629 | Saturday, 15th March 2008, 05:00 am | 3184 | 60% | hard |
275 | Balanced Sculptures | Let us define a balanced sculpture of order $n$ as follows:
A polyominoAn arrangement of identical squares connected through shared edges; holes are allowed. made up of $n + 1$ tiles known as the blocks ($n$ tiles) and the plinth (remaining tile);
the plinth has its centre at position ($x = 0, y = 0$);
the blocks have ... | Let us define a balanced sculpture of order $n$ as follows:
A polyominoAn arrangement of identical squares connected through shared edges; holes are allowed. made up of $n + 1$ tiles known as the blocks ($n$ tiles) and the plinth (remaining tile);
the plinth has its centre at position ($x = 0, y = 0$);
the blocks have ... | <p>Let us define a <dfn>balanced sculpture</dfn> of order $n$ as follows:
</p><ul><li>A <strong class="tooltip">polyomino<span class="tooltiptext">An arrangement of identical squares connected through shared edges; holes are allowed.</span></strong> made up of $n + 1$ tiles known as the <dfn>blocks</dfn> ($n$ tiles)<br... | 15030564 | Friday, 22nd January 2010, 05:00 pm | 684 | 85% | hard |
332 | Spherical Triangles | A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.
Let $C(r)$ be the sphere with the centre $(0,0,0)$ and radius $r$.
Let $Z(r)$ be the set of points on the surface of $C(r)$ with integer coordinates.
Let $T(r)$ be the set of spheric... | A spherical triangle is a figure formed on the surface of a sphere by three great circular arcs intersecting pairwise in three vertices.
Let $C(r)$ be the sphere with the centre $(0,0,0)$ and radius $r$.
Let $Z(r)$ be the set of points on the surface of $C(r)$ with integer coordinates.
Let $T(r)$ be the set of spheric... | <p>A <strong>spherical triangle</strong> is a figure formed on the surface of a sphere by three <strong>great circular arcs</strong> intersecting pairwise in three vertices.</p>
<div align="center"><img alt="0332_spherical.jpg" class="dark_img" src="resources/images/0332_spherical.jpg?1678992054"/></div>
<p>Let $C(r)$ ... | 2717.751525 | Sunday, 10th April 2011, 10:00 am | 682 | 50% | medium |
279 | Triangles with Integral Sides and an Integral Angle | How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed $10^8$? | How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed $10^8$? | <p>
How many triangles are there with integral sides, at least one integral angle (measured in degrees), and a perimeter that does not exceed $10^8$?
</p> | 416577688 | Saturday, 20th February 2010, 09:00 am | 840 | 60% | hard |
269 | Polynomials with at Least One Integer Root | A root or zero of a polynomial $P(x)$ is a solution to the equation $P(x) = 0$.
Define $P_n$ as the polynomial whose coefficients are the digits of $n$.
For example, $P_{5703}(x) = 5x^3 + 7x^2 + 3$.
We can see that:$P_n(0)$ is the last digit of $n$,
$P_n(1)$ is the sum of the digits of $n$,
$P_n(10)$ is $n$ itself.Def... | A root or zero of a polynomial $P(x)$ is a solution to the equation $P(x) = 0$.
Define $P_n$ as the polynomial whose coefficients are the digits of $n$.
For example, $P_{5703}(x) = 5x^3 + 7x^2 + 3$.
We can see that:$P_n(0)$ is the last digit of $n$,
$P_n(1)$ is the sum of the digits of $n$,
$P_n(10)$ is $n$ itself.Def... | <p>A root or zero of a polynomial $P(x)$ is a solution to the equation $P(x) = 0$. <br/>
Define $P_n$ as the polynomial whose coefficients are the digits of $n$.<br/>
For example, $P_{5703}(x) = 5x^3 + 7x^2 + 3$.</p>
<p>We can see that:</p><ul><li>$P_n(0)$ is the last digit of $n$,</li>
<li>$P_n(1)$ is the sum of the d... | 1311109198529286 | Saturday, 19th December 2009, 09:00 am | 789 | 80% | hard |
45 | Triangular, Pentagonal, and Hexagonal | Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle
$T_n=n(n+1)/2$
$1, 3, 6, 10, 15, \dots$
Pentagonal
$P_n=n(3n - 1)/2$
$1, 5, 12, 22, 35, \dots$
Hexagonal
$H_n=n(2n - 1)$
$1, 6, 15, 28, 45, \dots$
It can be verified that $T_{285} = P_{165} = H_{143} = 40755$.
Find... | Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:
Triangle
$T_n=n(n+1)/2$
$1, 3, 6, 10, 15, \dots$
Pentagonal
$P_n=n(3n - 1)/2$
$1, 5, 12, 22, 35, \dots$
Hexagonal
$H_n=n(2n - 1)$
$1, 6, 15, 28, 45, \dots$
It can be verified that $T_{285} = P_{165} = H_{143} = 40755$.
Find... | <p>Triangle, pentagonal, and hexagonal numbers are generated by the following formulae:</p>
<table><tr><td>Triangle</td>
<td> </td>
<td>$T_n=n(n+1)/2$</td>
<td> </td>
<td>$1, 3, 6, 10, 15, \dots$</td>
</tr><tr><td>Pentagonal</td>
<td> </td>
<td>$P_n=n(3n - 1)/2$</td>
<td> </td>
<td>$1, 5, 12, 22, 35, \dots$</td>
</tr><... | 1533776805 | Friday, 6th June 2003, 06:00 pm | 77303 | 5% | easy |
798 | Card Stacking Game | Two players play a game with a deck of cards which contains $s$ suits with each suit containing $n$ cards numbered from $1$ to $n$.
Before the game starts, a set of cards (which may be empty) is picked from the deck and placed face-up on the table, with no overlap. These are called the visible cards.
The players then... | Two players play a game with a deck of cards which contains $s$ suits with each suit containing $n$ cards numbered from $1$ to $n$.
Before the game starts, a set of cards (which may be empty) is picked from the deck and placed face-up on the table, with no overlap. These are called the visible cards.
The players then... | <p>
Two players play a game with a deck of cards which contains $s$ suits with each suit containing $n$ cards numbered from $1$ to $n$.</p>
<p>
Before the game starts, a set of cards (which may be empty) is picked from the deck and placed face-up on the table, with no overlap. These are called the visible cards.</p>
<p... | 132996198 | Saturday, 14th May 2022, 11:00 pm | 138 | 100% | hard |
815 | Group by Value | A pack of cards contains $4n$ cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and placed in piles of equal value. If the card has the same value as any pile it is placed in that pile. If there is no pile of that value then it begins a new pile. When a pile has four ... | A pack of cards contains $4n$ cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and placed in piles of equal value. If the card has the same value as any pile it is placed in that pile. If there is no pile of that value then it begins a new pile. When a pile has four ... | <p>
A pack of cards contains $4n$ cards with four identical cards of each value. The pack is shuffled and cards are dealt one at a time and placed in piles of equal value. If the card has the same value as any pile it is placed in that pile. If there is no pile of that value then it begins a new pile. When a pile has f... | 54.12691621 | Sunday, 6th November 2022, 01:00 am | 470 | 25% | easy |
477 | Number Sequence Game | The number sequence game starts with a sequence $S$ of $N$ numbers written on a line.
Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.
A player's own score is (determined by) the sum of all the numbers that player ha... | The number sequence game starts with a sequence $S$ of $N$ numbers written on a line.
Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.
A player's own score is (determined by) the sum of all the numbers that player ha... | <p>The number sequence game starts with a sequence $S$ of $N$ numbers written on a line.</p>
<p>Two players alternate turns. The players on their respective turns must select and remove either the first or the last number remaining in the sequence.</p>
<p>A player's own score is (determined by) the sum of all the numbe... | 25044905874565165 | Saturday, 23rd August 2014, 04:00 pm | 287 | 65% | hard |
220 | Heighway Dragon | Let D0 be the two-letter string "Fa". For n≥1, derive Dn from Dn-1 by the string-rewriting rules:
"a" → "aRbFR"
"b" → "LFaLb"
Thus, D0 = "Fa", D1 = "FaRbFR", D2 = "FaRbFRRLFaLbFR", and so on.
These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" ... | Let D0 be the two-letter string "Fa". For n≥1, derive Dn from Dn-1 by the string-rewriting rules:
"a" → "aRbFR"
"b" → "LFaLb"
Thus, D0 = "Fa", D1 = "FaRbFR", D2 = "FaRbFRRLFaLbFR", and so on.
These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" ... | <p>Let <b><i>D</i></b><sub>0</sub> be the two-letter string "Fa". For n≥1, derive <b><i>D</i></b><sub>n</sub> from <b><i>D</i></b><sub>n-1</sub> by the string-rewriting rules:</p>
<p style="margin-left:40px;">"a" → "aRbFR"<br>
"b" → "LFaLb"</br></p>
<p>Thus, <b><i>D</i></b><sub>0</sub> = "Fa", <b><i>D</i></b><sub>1</s... | 139776,963904 | Saturday, 6th December 2008, 09:00 am | 2379 | 55% | medium |
471 | Triangle Inscribed in Ellipse | The triangle $\triangle ABC$ is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, $0 \lt 2b \lt a$, $a$ and $b$ integers.
Let $r(a, b)$ be the radius of the incircle of $\triangle ABC$ when the incircle has center $(2b, 0)$ and $A$ has coordinates $\left( \frac a 2, \frac {\sqrt 3} 2 b\... | The triangle $\triangle ABC$ is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, $0 \lt 2b \lt a$, $a$ and $b$ integers.
Let $r(a, b)$ be the radius of the incircle of $\triangle ABC$ when the incircle has center $(2b, 0)$ and $A$ has coordinates $\left( \frac a 2, \frac {\sqrt 3} 2 b\... | <p>The triangle $\triangle ABC$ is inscribed in an ellipse with equation $\frac {x^2} {a^2} + \frac {y^2} {b^2} = 1$, $0 \lt 2b \lt a$, $a$ and $b$ integers.</p>
<p>Let $r(a, b)$ be the radius of the incircle of $\triangle ABC$ when the incircle has center $(2b, 0)$ and $A$ has coordinates $\left( \frac a 2, \frac {\sq... | 1.895093981e31 | Saturday, 10th May 2014, 10:00 pm | 239 | 75% | hard |
13 | Large Sum | Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.
37107287533902102798797998220837590246510135740250
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
2306758820753934617117198... | Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.
37107287533902102798797998220837590246510135740250
46376937677490009712648124896970078050417018260538
74324986199524741059474233309513058123726617309629
91942213363574161572522430563301811072406154908250
2306758820753934617117198... | <p>Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.</p>
<div class="monospace center">
37107287533902102798797998220837590246510135740250<br/>
46376937677490009712648124896970078050417018260538<br/>
74324986199524741059474233309513058123726617309629<br/>
919422133635741615725224... | 5537376230 | Friday, 22nd March 2002, 06:00 pm | 243707 | 5% | easy |
57 | Square Root Convergents | It is possible to show that the square root of two can be expressed as an infinite continued fraction.
$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$
By expanding this for the first four iterations, we get:
$1 + \frac 1 2 = \frac 32 = 1.5$
$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$
$1 + \frac 1 {2 + \... | It is possible to show that the square root of two can be expressed as an infinite continued fraction.
$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$
By expanding this for the first four iterations, we get:
$1 + \frac 1 2 = \frac 32 = 1.5$
$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$
$1 + \frac 1 {2 + \... | <p>It is possible to show that the square root of two can be expressed as an infinite continued fraction.</p>
<p class="center">$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$</p>
<p>By expanding this for the first four iterations, we get:</p>
<p>$1 + \frac 1 2 = \frac 32 = 1.5$<br/>
$1 + \frac 1 {2 + \frac... | 153 | Friday, 21st November 2003, 06:00 pm | 45920 | 5% | easy |
730 | Shifted Pythagorean Triples | For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a $k$-shifted Pythagorean triple if $$p^2 + q^2 + k = r^2$$
$(p, q, r)$ is said to be primitive if $\gcd(p, q, r)=1$.
Let $P_k(n)$ be the number of primitive $k$-shifted Pythagorean triples such that $1 \le p \le q \le r$ and $p + q... | For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a $k$-shifted Pythagorean triple if $$p^2 + q^2 + k = r^2$$
$(p, q, r)$ is said to be primitive if $\gcd(p, q, r)=1$.
Let $P_k(n)$ be the number of primitive $k$-shifted Pythagorean triples such that $1 \le p \le q \le r$ and $p + q... | <p>
For a non-negative integer $k$, the triple $(p,q,r)$ of positive integers is called a <dfn>$k$-shifted Pythagorean triple</dfn> if $$p^2 + q^2 + k = r^2$$
</p>
<p>
$(p, q, r)$ is said to be primitive if $\gcd(p, q, r)=1$.
</p>
<p>
Let $P_k(n)$ be the number of primitive $k$-shifted Pythagorean triples such that $1 ... | 1315965924 | Sunday, 18th October 2020, 08:00 am | 197 | 65% | hard |
517 | A Real Recursion | For every real number $a \gt 1$ is given the sequence $g_a$ by:
$g_{a}(x)=1$ for $x \lt a$
$g_{a}(x)=g_{a}(x-1)+g_a(x-a)$ for $x \ge a$
$G(n)=g_{\sqrt {n}}(n)$
$G(90)=7564511$.
Find $\sum G(p)$ for $p$ prime and $10000000 \lt p \lt 10010000$
Give your answer modulo $1000000007$. | For every real number $a \gt 1$ is given the sequence $g_a$ by:
$g_{a}(x)=1$ for $x \lt a$
$g_{a}(x)=g_{a}(x-1)+g_a(x-a)$ for $x \ge a$
$G(n)=g_{\sqrt {n}}(n)$
$G(90)=7564511$.
Find $\sum G(p)$ for $p$ prime and $10000000 \lt p \lt 10010000$
Give your answer modulo $1000000007$. | <p>
For every real number $a \gt 1$ is given the sequence $g_a$ by:<br/>
$g_{a}(x)=1$ for $x \lt a$<br/>
$g_{a}(x)=g_{a}(x-1)+g_a(x-a)$ for $x \ge a$<br/>
$G(n)=g_{\sqrt {n}}(n)$<br/>
$G(90)=7564511$.</p>
<p>
Find $\sum G(p)$ for $p$ prime and $10000000 \lt p \lt 10010000$<br/>
Give your answer modulo $1000000007$.
</... | 581468882 | Saturday, 23rd May 2015, 01:00 pm | 500 | 45% | medium |
647 | Linear Transformations of Polygonal Numbers | It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$. For example $F_3(100) = 184$.
Polygonal numbers are generalisations o... | It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$. For example $F_3(100) = 184$.
Polygonal numbers are generalisations o... | <p>
It is possible to find positive integers $A$ and $B$ such that given any triangular number, $T_n$, then $AT_n +B$ is always a triangular number. We define $F_3(N)$ to be the sum of $(A+B)$ over all such possible pairs $(A,B)$ with $\max(A,B)\le N$. For example $F_3(100) = 184$.
</p>
<p>
Polygonal numbers are genera... | 563132994232918611 | Sunday, 16th December 2018, 07:00 am | 483 | 30% | easy |
226 | A Scoop of Blancmange | The blancmange curve is the set of points $(x, y)$ such that $0 \le x \le 1$ and $y = \sum \limits_{n = 0}^{\infty} {\dfrac{s(2^n x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.
The area under the blancmange curve is equal to ½, shown in pink in the diagram below.
Let $C$ be the circle with ... | The blancmange curve is the set of points $(x, y)$ such that $0 \le x \le 1$ and $y = \sum \limits_{n = 0}^{\infty} {\dfrac{s(2^n x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.
The area under the blancmange curve is equal to ½, shown in pink in the diagram below.
Let $C$ be the circle with ... | <p>The <strong>blancmange curve</strong> is the set of points $(x, y)$ such that $0 \le x \le 1$ and $y = \sum \limits_{n = 0}^{\infty} {\dfrac{s(2^n x)}{2^n}}$, where $s(x)$ is the distance from $x$ to the nearest integer.</p>
<p>The area under the blancmange curve is equal to ½, shown in pink in the diagram below.</p... | 0.11316017 | Friday, 2nd January 2009, 09:00 pm | 1973 | 65% | hard |
678 | Fermat-like Equations | If a triple of positive integers $(a, b, c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a, b, c)$ satisfies $a^e+b^e=c^e$ when $e \ge 3$ (Fermat's Last Theorem). However, if the exponents of the left-hand side and right-hand side differ, this is not true. For example, $3^3+6^3=3^5$.
Let ... | If a triple of positive integers $(a, b, c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a, b, c)$ satisfies $a^e+b^e=c^e$ when $e \ge 3$ (Fermat's Last Theorem). However, if the exponents of the left-hand side and right-hand side differ, this is not true. For example, $3^3+6^3=3^5$.
Let ... | <p>If a triple of positive integers $(a, b, c)$ satisfies $a^2+b^2=c^2$, it is called a Pythagorean triple. No triple $(a, b, c)$ satisfies $a^e+b^e=c^e$ when $e \ge 3$ (Fermat's Last Theorem). However, if the exponents of the left-hand side and right-hand side differ, this is not true. For example, $3^3+6^3=3^5$.
</p... | 1986065 | Saturday, 7th September 2019, 10:00 pm | 267 | 55% | medium |
425 | Prime Connection | Two positive numbers $A$ and $B$ are said to be connected (denoted by "$A \leftrightarrow B$") if one of these conditions holds:
(1) $A$ and $B$ have the same length and differ in exactly one digit; for example, $123 \leftrightarrow 173$.
(2) Adding one digit to the left of $A$ (or $B$) makes $B$ (or $A$); for example,... | Two positive numbers $A$ and $B$ are said to be connected (denoted by "$A \leftrightarrow B$") if one of these conditions holds:
(1) $A$ and $B$ have the same length and differ in exactly one digit; for example, $123 \leftrightarrow 173$.
(2) Adding one digit to the left of $A$ (or $B$) makes $B$ (or $A$); for example,... | <p>
Two positive numbers $A$ and $B$ are said to be <dfn>connected</dfn> (denoted by "$A \leftrightarrow B$") if one of these conditions holds:<br/>
(1) $A$ and $B$ have the same length and differ in exactly one digit; for example, $123 \leftrightarrow 173$.<br/>
(2) Adding one digit to the left of $A$ (or $B$) makes $... | 46479497324 | Saturday, 27th April 2013, 04:00 pm | 1622 | 25% | easy |
305 | Reflexive Position | Let's call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from $1$) written down in base $10$.
Thus, $S = 1234567891011121314151617181920212223242\cdots$
It's easy to see that any number will show up an infinite number of times in $S$.
Let's call $f(n)$ the sta... | Let's call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from $1$) written down in base $10$.
Thus, $S = 1234567891011121314151617181920212223242\cdots$
It's easy to see that any number will show up an infinite number of times in $S$.
Let's call $f(n)$ the sta... | <p>
Let's call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from $1$) written down in base $10$.<br/>
Thus, $S = 1234567891011121314151617181920212223242\cdots$
</p>
<p>
It's easy to see that any number will show up an infinite number of times in $S$.
</p>
<p>
Le... | 18174995535140 | Sunday, 10th October 2010, 04:00 am | 688 | 60% | hard |
901 | Well Drilling | A driller drills for water. At each iteration the driller chooses a depth $d$ (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby.
Drilling to depth $d$... | A driller drills for water. At each iteration the driller chooses a depth $d$ (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby.
Drilling to depth $d$... | <p>A driller drills for water. At each iteration the driller chooses a depth $d$ (a positive real number), drills to this depth and then checks if water was found. If so, the process terminates. Otherwise, a new depth is chosen and a new drilling starts from the ground level in a new location nearby.</p>
<p>Drilling to... | 2.364497769 | Sunday, 21st July 2024, 08:00 am | 431 | 25% | easy |
844 | $k$-Markov Numbers | Consider positive integer solutions to
$a^2+b^2+c^2 = 3abc$
For example, $(1,5,13)$ is a solution. We define a 3-Markov number to be any part of a solution, so $1$, $5$ and $13$ are all 3-Markov numbers. Adding distinct 3-Markov numbers $\le 10^3$ would give $2797$.
Now we define a $k$-Markov number to be a positive in... | Consider positive integer solutions to
$a^2+b^2+c^2 = 3abc$
For example, $(1,5,13)$ is a solution. We define a 3-Markov number to be any part of a solution, so $1$, $5$ and $13$ are all 3-Markov numbers. Adding distinct 3-Markov numbers $\le 10^3$ would give $2797$.
Now we define a $k$-Markov number to be a positive in... | <p>Consider positive integer solutions to</p>
<center>$a^2+b^2+c^2 = 3abc$</center>
<p>For example, $(1,5,13)$ is a solution. We define a 3-Markov number to be any part of a solution, so $1$, $5$ and $13$ are all 3-Markov numbers. Adding distinct 3-Markov numbers $\le 10^3$ would give $2797$.</p>
<p>Now we define a $k$... | 101805206 | Saturday, 20th May 2023, 02:00 pm | 219 | 40% | medium |
764 | Asymmetric Diophantine Equation | Consider the following Diophantine equation:
$$16x^2+y^4=z^2$$
where $x$, $y$ and $z$ are positive integers.
Let $S(N) = \displaystyle{\sum(x+y+z)}$ where the sum is over all solutions $(x,y,z)$ such that $1 \leq x,y,z \leq N$ and $\gcd(x,y,z)=1$.
For $N=100$, there are only two such solutions: $(3,4,20)$ and $(10... | Consider the following Diophantine equation:
$$16x^2+y^4=z^2$$
where $x$, $y$ and $z$ are positive integers.
Let $S(N) = \displaystyle{\sum(x+y+z)}$ where the sum is over all solutions $(x,y,z)$ such that $1 \leq x,y,z \leq N$ and $\gcd(x,y,z)=1$.
For $N=100$, there are only two such solutions: $(3,4,20)$ and $(10... | <p>
Consider the following Diophantine equation:
$$16x^2+y^4=z^2$$
where $x$, $y$ and $z$ are positive integers.
</p>
<p>
Let $S(N) = \displaystyle{\sum(x+y+z)}$ where the sum is over all solutions $(x,y,z)$ such that $1 \leq x,y,z \leq N$ and $\gcd(x,y,z)=1$.
</p>
<p>
For $N=100$, there are only two such solutions: $... | 255228881 | Saturday, 11th September 2021, 05:00 pm | 395 | 40% | medium |
801 | $x^y \equiv y^x$ | The positive integral solutions of the equation $x^y=y^x$ are $(2,4)$, $(4,2)$ and $(k,k)$ for all $k > 0$.
For a given positive integer $n$, let $f(n)$ be the number of integral values $0 < x,y \leq n^2-n$ such that
$$x^y\equiv y^x \pmod n.$$
For example, $f(5)=104$ and $f(97)=1614336$.
Let $S(M,N)=\sum f(p)$ where th... | The positive integral solutions of the equation $x^y=y^x$ are $(2,4)$, $(4,2)$ and $(k,k)$ for all $k > 0$.
For a given positive integer $n$, let $f(n)$ be the number of integral values $0 < x,y \leq n^2-n$ such that
$$x^y\equiv y^x \pmod n.$$
For example, $f(5)=104$ and $f(97)=1614336$.
Let $S(M,N)=\sum f(p)$ where th... | <p>The positive integral solutions of the equation $x^y=y^x$ are $(2,4)$, $(4,2)$ and $(k,k)$ for all $k > 0$.</p>
<p>For a given positive integer $n$, let $f(n)$ be the number of integral values $0 < x,y \leq n^2-n$ such that
$$x^y\equiv y^x \pmod n.$$
For example, $f(5)=104$ and $f(97)=1614336$.</p>
<p>Let $S(M... | 638129754 | Sunday, 5th June 2022, 08:00 am | 300 | 50% | medium |
811 | Bitwise Recursion | Let $b(n)$ be the largest power of 2 that divides $n$. For example $b(24) = 8$.
Define the recursive function:
\begin{align*}
\begin{split}
A(0) &= 1\\
A(2n) &= 3A(n) + 5A\big(2n - b(n)\big) \qquad n \gt 0\\
A(2n+1) &= A(n)
\end{split}
\end{align*}
and let $H(t,r) = A\big((2^t+1)^r\big)$.
You are given $H(3,2) = A(8... | Let $b(n)$ be the largest power of 2 that divides $n$. For example $b(24) = 8$.
Define the recursive function:
\begin{align*}
\begin{split}
A(0) &= 1\\
A(2n) &= 3A(n) + 5A\big(2n - b(n)\big) \qquad n \gt 0\\
A(2n+1) &= A(n)
\end{split}
\end{align*}
and let $H(t,r) = A\big((2^t+1)^r\big)$.
You are given $H(3,2) = A(8... | <p>
Let $b(n)$ be the largest power of 2 that divides $n$. For example $b(24) = 8$.</p>
<p>
Define the recursive function:
\begin{align*}
\begin{split}
A(0) &= 1\\
A(2n) &= 3A(n) + 5A\big(2n - b(n)\big) \qquad n \gt 0\\
A(2n+1) &= A(n)
\end{split}
\end{align*}
and let $H(t,r) = A\big((2^t+1)^r\big)$.</p>
<... | 327287526 | Saturday, 8th October 2022, 02:00 pm | 244 | 45% | medium |
592 | Factorial Trailing Digits 2 | For any $N$, let $f(N)$ be the last twelve hexadecimal digits before the trailing zeroes in $N!$.
For example, the hexadecimal representation of $20!$ is 21C3677C82B40000,
so $f(20)$ is the digit sequence 21C3677C82B4.
Find $f(20!)$. Give your answer as twelve hexadecimal digits, using uppercase for the digits A to F. | For any $N$, let $f(N)$ be the last twelve hexadecimal digits before the trailing zeroes in $N!$.
For example, the hexadecimal representation of $20!$ is 21C3677C82B40000,
so $f(20)$ is the digit sequence 21C3677C82B4.
Find $f(20!)$. Give your answer as twelve hexadecimal digits, using uppercase for the digits A to F. | <p>For any $N$, let $f(N)$ be the last twelve hexadecimal digits before the trailing zeroes in $N!$.</p>
<p>For example, the hexadecimal representation of $20!$ is 21C3677C82B40000,<br/>
so $f(20)$ is the digit sequence 21C3677C82B4.</p>
<p>Find $f(20!)$. Give your answer as twelve hexadecimal digits, using uppercase f... | 13415DF2BE9C | Saturday, 25th February 2017, 04:00 pm | 321 | 60% | hard |
785 | Symmetric Diophantine Equation | Consider the following Diophantine equation:
$$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$
where $x$, $y$ and $z$ are positive integers.
Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1 \le x \le y \le z \le N$ and $\gcd(x, y, z) = 1$.
For $N = 10^2$, there are three such solutions... | Consider the following Diophantine equation:
$$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$
where $x$, $y$ and $z$ are positive integers.
Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1 \le x \le y \le z \le N$ and $\gcd(x, y, z) = 1$.
For $N = 10^2$, there are three such solutions... | <p>
Consider the following Diophantine equation:
$$15 (x^2 + y^2 + z^2) = 34 (xy + yz + zx)$$
where $x$, $y$ and $z$ are positive integers.
</p>
<p>
Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1 \le x \le y \le z \le N$ and $\gcd(x, y, z) = 1$.
</p>
<p>
For $N = 10^2$, there are th... | 29526986315080920 | Sunday, 13th February 2022, 07:00 am | 209 | 55% | medium |
555 | McCarthy 91 Function | The McCarthy 91 function is defined as follows:
$$
M_{91}(n) =
\begin{cases}
n - 10 & \text{if } n > 100 \\
M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100
\end{cases}
$$
We can generalize this definition by abstracting away the constants into new variables:
$$
M_{m,k,s}(n) =
\begin... | The McCarthy 91 function is defined as follows:
$$
M_{91}(n) =
\begin{cases}
n - 10 & \text{if } n > 100 \\
M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100
\end{cases}
$$
We can generalize this definition by abstracting away the constants into new variables:
$$
M_{m,k,s}(n) =
\begin... | <p>
The McCarthy 91 function is defined as follows:
$$
M_{91}(n) =
\begin{cases}
n - 10 & \text{if } n > 100 \\
M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100
\end{cases}
$$
</p>
<p>
We can generalize this definition by abstracting away the constants into new variables:
$$
M_{m... | 208517717451208352 | Sunday, 10th April 2016, 04:00 am | 781 | 30% | easy |
693 | Finite Sequence Generator | Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner:
$a_x = y$ is the first term,
$a_{z+1} = a_z^2 \bmod z$ for $z = x, x+1,x+2,\ldots$ and
the generation stops when a term becomes $0$ or $1$.
The number of terms in this sequence is denoted $l(x,y)$.
For example, with $x = 5$ a... | Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner:
$a_x = y$ is the first term,
$a_{z+1} = a_z^2 \bmod z$ for $z = x, x+1,x+2,\ldots$ and
the generation stops when a term becomes $0$ or $1$.
The number of terms in this sequence is denoted $l(x,y)$.
For example, with $x = 5$ a... | <p>Two positive integers $x$ and $y$ ($x > y$) can generate a sequence in the following manner:</p>
<ul>
<li>$a_x = y$ is the first term,</li>
<li>$a_{z+1} = a_z^2 \bmod z$ for $z = x, x+1,x+2,\ldots$ and</li>
<li>the generation stops when a term becomes $0$ or $1$.</li>
</ul>
<p>The number of terms in this sequence... | 699161 | Saturday, 14th December 2019, 04:00 pm | 310 | 40% | medium |
205 | Dice Game | Peter has nine four-sided (pyramidal) dice, each with faces numbered $1, 2, 3, 4$.
Colin has six six-sided (cubic) dice, each with faces numbered $1, 2, 3, 4, 5, 6$.
Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.
What is the probability that Pyr... | Peter has nine four-sided (pyramidal) dice, each with faces numbered $1, 2, 3, 4$.
Colin has six six-sided (cubic) dice, each with faces numbered $1, 2, 3, 4, 5, 6$.
Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.
What is the probability that Pyr... | <p>Peter has nine four-sided (pyramidal) dice, each with faces numbered $1, 2, 3, 4$.<br/>
Colin has six six-sided (cubic) dice, each with faces numbered $1, 2, 3, 4, 5, 6$.</p>
<p>Peter and Colin roll their dice and compare totals: the highest total wins. The result is a draw if the totals are equal.</p>
<p>What is th... | 0.5731441 | Saturday, 6th September 2008, 02:00 pm | 16153 | 15% | easy |
218 | Perfect Right-angled Triangles | Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$.
The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and $28$.
Moreover it is a primitive right angled triangle as $\gcd(a,b)=1$ and $\gcd(b,c)=1$.
Also $c$ is a perfect square.
We will call a right angled triangle perfe... | Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$.
The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and $28$.
Moreover it is a primitive right angled triangle as $\gcd(a,b)=1$ and $\gcd(b,c)=1$.
Also $c$ is a perfect square.
We will call a right angled triangle perfe... | <p>Consider the right angled triangle with sides $a=7$, $b=24$ and $c=25$.
The area of this triangle is $84$, which is divisible by the perfect numbers $6$ and $28$.<br/>
Moreover it is a primitive right angled triangle as $\gcd(a,b)=1$ and $\gcd(b,c)=1$.<br/>
Also $c$ is a perfect square.</p>
<p>We will call a right a... | 0 | Saturday, 22nd November 2008, 01:00 am | 3325 | 55% | medium |
49 | Prime Permutations | The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit primes, exhibit... | The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit primes, exhibit... | <p>The arithmetic sequence, $1487, 4817, 8147$, in which each of the terms increases by $3330$, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the $4$-digit numbers are permutations of one another.</p>
<p>There are no arithmetic sequences made up of three $1$-, $2$-, or $3$-digit prime... | 296962999629 | Friday, 1st August 2003, 06:00 pm | 63593 | 5% | easy |
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