messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \\geq 0$. Suppose that there exists a fixed no... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{P}_1$ and $\\mathcal{P}_2$ be two parabolas with distinct directrices $\\ell_1$ and $\\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is kno... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "25) A planet orbits around a star S, as shown in the figure. The semi-major axis of the orbit is a. The perigee, namely the shortest distance between the planet and the star is 0... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function $f$ , defined on the set of ordered pairs of positive integers, satisfies the following properties: \\[f(x, x) = x,\\; f(x, y) = f(y, x), {\\rm \\ and\\ } (x+y)f(x, ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For how many ordered pairs of positive integers $(x,y),$ with $y<x\\le 100,$ are both $\\frac xy$ and $\\frac{x+1}{y+1}$ integers?",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the arc length of the curve, where x=t, y=t^2 and z=2*t^3/3.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the arc length of the curve defin... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 2,$ $|z_2| = 3,$ and $|z_3| = 4.$ Find the largest possible value of\n\\[|z_1 - z_2|^2 + |z_1 - z_3|^2 + |z_2 - z_3... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For how many ordered pairs $(x,y)$ of integers is it true that $0 < x < y < 10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC,$ let $D,$ $E,$ $F$ be the midpoints of $\\overline{BC},$ $\\overline{AC},$ $\\overline{AB},$ respectively. Let $P,$ $Q,$ $R$ be the midpoints of $\\overline{AD... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ peb... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit numbers and will have the property ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all $n$ for which the system with of equations can be solved in $\\mathbb{R}$ :\r\n\r\n\\[\\sum^{n}_{k=1} x_k = 27\\]\r\n\r\nand\r\n\r\n\\[\\prod^{n}_{k=1} x_k = \\l... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\Sigma$ be a finite set. For $x,y \\in \\Sigma^{\\ast}$ , define \\[x\\preceq y\\] if $x$ is a sub-string (**not necessarily contiguous**) of $y$ . For example, $ac ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "With all angles measured in degrees, the product $\\prod_{k=1}^{45} \\csc^2(2k-1)^\\circ=m^n$ , where $m$ and $n$ are integers greater than 1. Find $m+n$ .",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "d. If Carla's stopwatch measures an elapsed time $\\tau$, what does Clock B measure in CIRF?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to figure out ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(b) Compute the electric potential both inside and outside the sphere.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to compute the electric potential bo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$ . Let $\\theta = \\arg \\left(\\tfrac{w-z}{z}\\right)$ . The maximum possible value of $\\tan^2 \\theta$ can... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two pl... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A trinitro compound, 1,3,5-tris-(4-nitrophenyl)benzene, on complete reaction with an excess of $\\mathrm{Sn} / \\mathrm{HCl}$ gives a major product, which on treatment with an ex... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Astronomers are searching for exoplanets around two stars with exactly the same masses. Using the RV method, they detected one planet around each star, both with minimum masses s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$ . Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(d) The one-dimensional diffusion equation (also called the \"heat equation\") is given (for a free particle) by\n\n$$\n\\frac{\\partial \\psi}{\\partial t}=a \\frac{\\partial^{2... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ n \\geq 2$ be a positive integer and $ \\lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a m... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its nonempty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, b... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the le... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such away that the point $(0, 0)$ is labeled by $0$ , and for every $x, y \\geq 0$ , ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonal... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \\dots, k$ . It is known that for $i = 1, 2, \\dots, k$ , Part $i$ contains $i$ multiple choice quest... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let triangle $ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\\overline{AB}$ with $D$ between $A$ and $E$ such that $\\overline{CD}$ and $\\overl... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes.... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ .",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the remainder when a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n\\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m \\circ n = \\frac{m+n}{mn+4}$. Compute $((\\cdots((2005 \\circ 2004) \\circ 2003) \\circ \\cdots \\circ 1) \\circ 0)$.",
"role": "user"
},
{
"content": "<think... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For the integer $n>1$ , define $D(n)=\\{ a-b\\mid ab=n, a>b>0, a,b\\in\\mathbb{N} \\}$ . Prove that for any integer $k>1$ , there exists pairwise distinct positive integers $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A molecule consists of three atoms with spin ½ each. The spin vector of the molecule can be written as $s = \\sigma/2$, where $\\sigma$ are Pauli matrices. The interaction Hamilt... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $I(R)=\\iint_{x^2+y^2 \\leq R^2}(\\frac{1+2 x^2}{1+x^4+6x^2y^2+y^4}-\\frac{1+y^2}{2+x^4+y^4}) dx dy$. What is the limit of $I(R)$ as $R$ goes to infinity?",
"role": "user... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ , $AD$ is parallel to $BC$ . Knowing that $AB=AD+BC$ , prove that the bisector of $\\angle A$ also bisects $CD$ .",
"role": "user"
},
{
"c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Trapezoid $ABCD^{}_{}$ has sides $AB=92^{}_{}$ , $BC=50^{}_{}$ , $CD=19^{}_{}$ , and $AD=70^{}_{}$ , with $AB^{}_{}$ parallel to $CD^{}_{}$ . A circle with center $P^{}_{}$ on $A... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a resistor made from a hollow cylinder of carbon as shown below. The inner radius of the cylinder is $R_i=0.2$mm and the outer radius is $R_o=0.3$mm. The length of the r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Exactly 2/5th of the children in a certain class are girls. If there are 100 boys in the class, how many girls are in the class?",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the sum of all numbers of the form $\\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive divisors of $1000.$ What is the greatest integer that does not exce... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The vector $\\begin{pmatrix} 1 \\\\ 2 \\\\ 2 \\end{pmatrix}$ is rotated $90^\\circ$ about the origin. During the rotation, it passes through the $x$-axis. Find the resulting ve... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathcal{S}$ be the set $\\lbrace1,2,3,\\ldots,10\\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{S}$ . (Disjoint sets are defined as... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For integers $k\\ (0\\leq k\\leq 5)$ , positive numbers $m,\\ n$ and real numbers $a,\\ b$ , let $f(k)=\\int_{-\\pi}^{\\pi}(\\sin kx-a\\sin mx-b\\sin nx)^{2}\\ dx$ , \r $p(k... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Polyhedron $ABCDEFG$ has six faces. Face $ABCD$ is a square with $AB = 12;$ face $ABFG$ is a trapezoid with $\\overline{AB}$ parallel to $\\overline{GF},$ $BF = AG = 8,$ and $GF... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After wait... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find continuous functions $x(t),\\ y(t)$ such that \r\n\r $\\ \\ \\ \\ \\ \\ \\ \\ \\ x(t)=1+\\int_{0}^{t}e^{-2(t-s)}x(s)ds$ \r\n\r $\\ \\ \\ \\ \\ \\ \\ \\ \\ y(t)=\\int_{0}^{... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a real number $x,$ let $\\lfloor x \\rfloor$ denote the greatest integer less than or equal to $x.$ For a certain integer $k,$ there are exactly $70$ positive integers $n_{... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "It is given that $\\log_{6}a + \\log_{6}b + \\log_{6}c = 6$, where $a$, $b$, and $c$ are positive integers that form an increasing geometric sequence and $b - a$ is the square of... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the minimum value of $\\int_0^1 \\{\\sqrt{x}-(a+bx)\\}^2dx$ .\n\nPlease solve the problem without using partial differentiation for those who don't learn it.\n\n1961 Waseda... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC$ is a right triangle with $AC = 7,$ $BC = 24,$ and right angle at $C.$ Point $M$ is the midpoint of $AB,$ and $D$ is on the same side of line $AB$ as $C$ so that $A... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCD$ be a parallelogram . Extend $\\overline{DA}$ through $A$ to a point $P,$ and let $\\overline{PC}$ meet $\\overline{AB}$ at $Q$ and $\\overline{DB}$ at $R.$ Given that... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The shortest distances between an interior diagonal of a rectangular parallelepiped , $P$ , and the edges it does not meet are $2\\sqrt{5}$ , $\\frac{30}{\\sqrt{13}}$ , and $\\fr... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7$ . Two spheres with equal r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a file with a size of 350 Kbytes storing in a web server. Client A sends a request to the server to retrieve the file from a remote location. It is known that the link c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $ P_1(x)\\equal{}\\frac{d}{dx}(x^2\\minus{}1),\\ P_2(x)\\equal{}\\frac{d^2}{dx^2}(x^2\\minus{}1)^2,\\ P_3(x)\\equal{}\\frac{d^3}{dx^3}(x^2\\minus{}1)^3$ .\r\n\r\nFi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "B.4 If $\\Delta E_{r m s}=5.54 \\times 10^{-17} \\mathrm{~J}$, calculate the rms speed of the Be nuclei, $\\mathrm{V}_{\\mathrm{Be}}$, and hence estimate $T_{\\mathrm{c}}$. (Hint... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the area of the region enclosed by the graph of $|x-60|+|y|=\\left|\\frac{x}{4}\\right|.$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The $52$ cards in a deck are numbered $1, 2, \\cdots, 52$ . Alex, Blair, Corey, and Dylan each pick a card from the deck randomly and without replacement. The two people with low... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For some integer $m$ , the polynomial $x^3 - 2011x + m$ has the three integer roots $a$ , $b$ , and $c$ . Find $|a| + |b| + |c|$ .",
"role": "user"
},
{
"content": "... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$ . Find the sum of all positive integers $n$ for which $\\left |\\sum_{k=1}^n\\log_{10}f(k)\\right|=1.$",
"role": "user"
},
{
"co... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The points $A$ , $B$ and $C$ lie on the surface of a sphere with center $O$ and radius $20$ . It is given that $AB=13$ , $BC=14$ , $CA=15$ , and that the distance from $O$ to tri... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "3.4\n\nCalculate the energy of the ion $\\mathrm{A}^{(\\mathrm{Z}-2)+}$ taken to be in the ground state, using the same principle as the calculation of the energy of $\\mathrm{A}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A particle of mass m moves in a 1D potential given by V(x) = -α δ(x), where δ(x) is the usual Dirac delta function, and α is some positive constant. The particle is bound. Find t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be a subset of $\\mathbb{Z}/n\\mathbb{Z}$ with at most $\\frac{\\ln(n)}{100}$ elements. \nDefine $f(r)=\\sum_{s\\in A} e^{\\dfrac{2 \\pi i r s}{n}}$ . Show that fo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2. The potential due to the line charge could be written as\n\n$$\nV(r)=f(r)+K,\n$$\n\nwhere $K$ is a constant. Determine $f(r)$.",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = \\|\\mathbf{b}\\| = 1$ and $\\|\\mathbf{c}\\| = 2.$ Find the maximum value of\n\\[\\|\\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$ , let $BC = 1000$ and $AD = 2008$ . Let $\\angle A = 37^\\circ$ , $\\angle D = 53^\\circ$ , and $M$ and $N$ be t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(t)$ be the cubic polynomial for $t$ such that $\\cos 3x=f(\\cos x)$ holds for all real number $x$ .\r\nEvaluate\r\n\r\n\\[\\int_0^1 \\{f(t)\\}^2 \\sqrt{1-t^2}dt\\]"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are real numbers $a, b, c,$ and $d$ such that $-20$ is a root of $x^3 + ax + b$ and $-21$ is a root of $x^3 + cx^2 + d.$ These two polynomials share a complex root $m + \\s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Positive integers $a$ , $b$ , $c$ , and $d$ satisfy $a > b > c > d$ , $a + b + c + d = 2010$ , and $a^2 - b^2 + c^2 - d^2 = 2010$ . Find the number of possible values of $a$ .",
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$ , $x_1$ , $\\dots$ , $x_{2011}$ such that \\[m^{x_0} = \\sum_{k = 1}^{2011} m^{x_k}.\\]"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ with $\\overline{BC}\\parallel\\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\\angle A = 37^\\circ$, $\\angle D = 53^\\circ$, and $M$ and $N$ be the m... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sequences of positive integers $1,a_2, a_3,...$ and $1,b_2, b_3,...$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_1 = a_2 = a_3 = 1.$ For $n > 3,$ let $a_n$ be the number of real numbers $x$ such that\n\\[x^4 - 2a_{n - 1} x^2 + a_{n - 2} a_{n - 3} = 0.\\]Compute the sum $a_1 + a_2 + ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sa... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .",
"role": "user"
},
{
"co... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many different positive values of $x$ will make this statement true: there are exactly $2$ positive two-digit multiples of $x$.",
"role": "user"
},
{
"content": "... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$ , point $D$ is on $\\overline{BC}$ with $CD=2$ and $DB=5$ , point $E$ is on $\\overline{AC}$ with $CE=1$ and $EA=3$ , $AB=8$ , and $\\overline{AD}$ and $\\overl... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\\overline{BC}$ and $\\overline{CD},$ respectively, so that $\\triangle AEF$ is equilateral. A square with vertex ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A researcher observes at his laboratory that a muon travels a distance of 1000m after its creation to its disintegration. If the average lifetime of a muon is $2 \\times 10^{-6}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The bromination of alkenes is a stereospecific reaction. However, 2-methoxybut-2-ene undergoes bromination in carbon tetrachloride to form a mixture of cis and trans-dihalides. R... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A function $f:[0,\\infty)\\to[0,\\infty)$ is integrable and $$ \\int_0^\\infty f(x)^2 dx<\\infty,\\quad \\int_0^\\infty xf(x) dx <\\infty $$ Prove the following inequality. ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of integers between 1 and $2^{40}$ whose binary expansions have exactly two 1's. If a number is chosen at random from $S,$ the probability that it is divisible... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a set $L$ of lines in general position in the plane (no two lines in $L$ are parallel, and no three lines are concurrent) and another line $\\ell$ , show that the tota... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Let $m/n$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le |v+w|,$ where $m$ and ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a,b,c,d$ be positive integers such that $a+c=20$ and $\\frac{a}{b}+\\frac{c}{d}<1$ . Find the maximum possible value of $\\frac{a}{b}+\\frac{c}{d}$ .",
"role": "us... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "During a quantum communication task of a quantum dialogue for a two-mode entangled squeezed continuous variable state under noiseless quantum channel such that the two modes gene... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$ , $|z_k z_{k+1}|=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.2.2. Calculate the numerical value of $\\Gamma$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to calculate the numerical value of Γ, but wait, Γ usual... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(d) Using parts (a), (b), and (c), show that the average power delivered to the resistor (or produced by the resistor) per frequency interval is\n\n$$\nP[f, f+d f] \\approx k T d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\Gamma$ denote a curve $y=y(x)$ which is in the first quadrant and let the point $(1,0)$ lie on it. Let the tangent to $\\Gamma$ at a point $P$ intersect the $y$-axis at $Y... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ f: Z \\to Z$ be such that $ f(1) \\equal{} 1, f(2) \\equal{} 20, f(\\minus{}4) \\equal{} \\minus{}4$ and $ f(x\\plus{}y) \\equal{} f(x) \\plus{}f(y)\\plus{}axy(x\\plus... |
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