messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let \\(n\\ge3\\) be a fixed integer, and let \\(\\alpha\\) be a fixed positive real number. There are \\(n\\) numbers written around a circle such that there is exactly one \\(1\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with inf... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 1... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The set of points $\\left(x_{1}, x_{2}, x_{3}, x_{4}\\right)$ in $\\mathbf{R}^{4}$ such that $x_{1} \\geq x_{2} \\geq x_{3} \\geq x_{4}$ is a cone (or hypercone, if you insist). I... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Equilateral triangle $\\triangle ABC$ is inscribed in circle $\\omega$ with radius $18.$ Circle $\\omega_A$ is tangent to sides $\\overline{AB}$ and $\\overline{AC}$ and is intern... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a set of size 3. How many collections $T$ of subsets of $S$ have the property that for any two subsets $U \\in T$ and $V \\in T$, both $U \\cap V$ and $U \\cup V$ are i... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Take the following differential equation:\n\\[3(3+x^2)\\frac{dx}{dt}=2(1+x^2)^2e^{-t^2}\\]\nIf $x(0)\\leq 1$ , prove that there exists $M>0$ such that $|x(t)|<M$ for all $t\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given complex number $z$, define sequence $z_{0}, z_{1}, z_{2}, \\ldots$ as $z_{0}=z$ and $z_{n+1}=2 z_{n}^{2}+2 z_{n}$ for $n \\geq 0$. Given that $z_{10}=2017$, find the minimum... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\\frac{1}{t_{(n-1)}}$ when $n$ is odd. ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Positive integers $ a$ , $ b$ , and $ c$ are chosen so that $ a<b<c$ , and the system of equations\r\n\\[ 2x\\plus{}y\\equal{}2003\\text{ and }y\\equal{}|x\\minus{}a|\\plus... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "(a) Does $\\sum_{i=1}^{p-1} \\frac{1}{i} \\equiv 0\\left(\\bmod p^{2}\\right)$ for all odd prime numbers $p$? (Note that $\\frac{1}{i}$ denotes the number such that $\\left.i \\cd... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A circle is inscribed in quadrilateral $ABCD$ , tangent to $\\overline{AB}$ at $P$ and to $\\overline{CD}$ at $Q$ . Given that $AP=19$ , $PB=26$ , $CQ=37$ , and $QD=23$ , find the... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The function $f(x)$ is of the form $a x^{2}+b x+c$ for some integers $a, b$, and $c$. Given that $\\{f(177883), f(348710), f(796921), f(858522)\\} = \\{1324754875645,1782225466694... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define $p(x) = 4x^{3}-2x^{2}-15x+9, q(x) = 12x^{3}+6x^{2}-7x+1$ . Show that each polynomial has just three distinct real roots. Let $A$ be the largest root of $p(x)$ and $B$... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A compound with the molecular formula C8H6O2 gives a positive DNP test, a positive silver mirror test, and also reacts with KMnO4 to give benzoic acid. Which one of the following ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $p(x,y) = a_0 + a_1x + a_2y + a_3x^2 + a_4xy + a_5y^2 + a_6x^3 + a_7x^2y + a_8xy^2 + a_9y^3$ . Suppose that $p(0,0) = p(1,0) = p( - 1,0) = p(0,1) = p(0, - 1) = p(1,1) = p(1, -... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of $E_{0}=m c^{2}$ and the... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $F(z)=\\frac{z+i}{z-i}$ for all complex numbers $z\\not= i$ , and let $z_n=F(z_{n-1})$ for all positive integers $n$ . Given that $z_0=\\frac 1{137}+i$ and $z_{2002}=a+bi$ , w... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $a_{i,j}\\enspace(\\forall\\enspace 1\\leq i\\leq n, 1\\leq j\\leq n)$ be $n^2$ real numbers such that $a_{i,j}+a_{j,i}=0\\enspace\\forall i, j$ (in particular, $a_{i,i... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Prove that $$ \\int^{\\frac\\pi3}_{\\frac\\pi6}\\frac u{\\sin u}du=\\frac83\\sum_{k=0}^\\infty\\frac{(-1)^k}{3^k(2k+1)^2}+\\frac{\\pi\\ln3}{3\\sqrt3}-\\frac{4C}3+\\frac\\pi6\\ln\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \\geq 1$ . How many possible values of t are there, if $a_{1998} = 0$ ?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $R$ ba a finite ring with the following property: for any $a,b\\in R$ there exists an element $c\\in R$ (depending on $a$ and $b$ ) such that $a^2+b^2=c^2$ .\nProve ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$ , $BC = 14$ , and $AD = 2\\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of th... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega = e^{2 \\pi i/1729}.$ Compute\n\\[\\sum_{k = 1}^{1728} \\frac{1}{1 + \\omega^k + \\omega^{2k} + \\omega^{3k}}.\\]",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A small point-like object is thrown horizontally off of a $50.0$ - $\\text{m}$ high building with an initial speed of $10.0 \\text{ m/s}$ . At any point along the trajectory t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define a sequence recursively by $t_1 = 20$ , $t_2 = 21$ , and \\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\] for all $n \\ge 3$ . Then $t_{2020}$ can be written as $\\frac{p}{q}$ , wh... |
[
{
"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": "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 si... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D^{}_{}$ be the difference... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Tom is chasing Jerry on the coordinate plane. Tom starts at $(x, y)$ and Jerry starts at $(0, 0)$ . Jerry moves to the right at $1$ unit per second. At each positive integer ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The Proca equation is given by\n\n\\partial_{\\mu}F^{\\mu\\nu}+m^{2}A^{\\nu}=0\n\nif \\mu,\\nu=0,…,n, what is the general solution of the motion equations A^{\\nu}?\nAnswer Choice... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A teacher was leading a class of four perfectly logical students. The teacher chose a set $S$ of four integers and gave a different number in $S$ to each student. Then the teacher... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "There are $N$ permutations $(a_1, a_2, \\dots, a_{30})$ of $1, 2, \\dots, 30$ such that for $m \\in \\{2,3,5\\}$ , $m$ divides $a_{n+m} - a_n$ for all integers $n$ with $1 \\le n ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The sum of the areas of all triangles whose vertices are also vertices of a 1 by 1 by 1 cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$"... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a set with six elements . Let $\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$ , not necessarily distinct, are chosen independently and at ra... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $n\\ge 1$ be a fixed integer. Calculate the distance $\\inf_{p,f}\\, \\max_{0\\le x\\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ w... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Bob’s Rice ID number has six digits, each a number from $1$ to $9$ , and any digit can be used any number of times. The ID number satifies the following property: the first two... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given a number $n\\in\\mathbb{Z}^+$ and let $S$ denotes the set $\\{0,1,2,...,2n+1\\}$ . Consider the function $f:\\mathbb{Z}\\times S\\to [0,1]$ satisfying two following c... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\\frac 32$ minutes and red for 1 minute. For which $v$... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The reaction of silver and oxygen to silver oxide Ag2O has a standard Gibbs free energy (deltaG) of -22 kJ/mol (25 centigrade, 101300 Pa). When placing 23.2 g of solid Ag2O in 1 L... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ have side lengths $AB=30$ , $BC=32$ , and $AC=34$ . Point $X$ lies in the interior of $\\overline{BC}$ , and points $I_1$ and $I_2$ are the incenters of $\\tr... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The root of the equation x = (1 / 2) + sin x by using the iteration method: x_{k+1} = 1/2 + sin(x_k), x_0 = 1 correct to o six decimals is x = 1.497300. Determine the number of it... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $a_{0}=-2, b_{0}=1$, and for $n \\geq 0$, let $$\\begin{aligned} & a_{n+1}=a_{n}+b_{n}+\\sqrt{a_{n}^{2}+b_{n}^{2}} \\\\ & b_{n+1}=a_{n}+b_{n}-\\sqrt{a_{n}^{2}+b_{n}^{2}} \\end... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose a parabola with the axis as the $ y$ axis, concave up and touches the graph $ y\\equal{}1\\minus{}|x|$ . Find the equation of the parabola such that the area of the reg... |
[
{
"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": "use... |
[
{
"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 ca... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $EFGH$ , $EFDC$ , and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$ , and let $A$ be the eighth vertex of the cube. Let $I$ , $J$ , and $K$ , be the poin... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as $1/i^2$. The total received signal at time $i$ is $Y_i = \\frac{1}{i}... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In this problem, we consider a simple model for a thermoacoustic device. The device uses heavily amplified sound to provide work for a pump that can then extract heat. Sound waves... |
[
{
"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 t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider an ultra high energy proton coming from deep space. This proton collide with a photon from the cosmic microwave background. This collision is head-on. During this process... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A,\\text{ }B,\\text{ }C$ , which can each be inscribed in ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the maximum value of $ x_{0}$ for which there exists a sequence $ x_{0},x_{1}\\cdots ,x_{1995}$ of positive reals with $ x_{0} \\equal{} x_{1995}$ , such that\r\n\\[ x_{... |
[
{
"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": "(1) Find the range of $a$ for which there exist two common tangent lines of the curve $y=\\frac{8}{27}x^3$ and the parabola $y=(x+a)^2$ other than the $x$ axis.\n\n(2) For... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "(a) A function $f:\\mathbb{Z} \\rightarrow \\mathbb{Z}$ is called $\\mathbb{Z}$ -good if $f(a^2+b)=f(b^2+a)$ for all $a, b \\in \\mathbb{Z}$ . What is the largest possible n... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The positive numbers $a_1, a_2,...$ satisfy $a_1 = 1$ and $(m+n)a_{m+n }\\le a_m +a_n$ for all positive integers $m$ and $n$ . Show that $\\frac{1}{a_{200}} > 4 \\cdot... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Simplify the product $$\\prod_{m=1}^{100} \\prod_{n=1}^{100} \\frac{x^{n+m}+x^{n+m+2}+x^{2 n+1}+x^{2 m+1}}{x^{2 n}+2 x^{n+m}+x^{2 m}}$$ Express your answer in terms of $x$.",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $R$ be an associative non-commutative ring and let $n>2$ be a fixed natural number. Assume that $x^n=x$ for all $x\\in R$ . Prove that $xy^{n-1}=y^{n-1}x$ holds for a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\\ge 3$ ,\n\\[a_n=\\frac12na_{n-1}+\\frac12n(n-1)a_{n-2}+(-1)^n\\left(1-\\frac{n}{2}\\right).\\]\nFind a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $f: \\mathbb{N} \\rightarrow \\mathbb{N}$ be a function satisfying the following conditions: (a) $f(1)=1$ (b) $f(a) \\leq f(b)$ whenever $a$ and $b$ are positive integers with... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the set\n\n\\[M = \\left\\{ \n\\begin{pmatrix}\na & b \nc & d\n\\end{pmatrix}\n\\in\\mathcal{M}_2(\\mathbb{C})\\ |\\ ab = cd\n\\right\\}.\\]\n\na) Give an example of matr... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\\{d_1,d_2,\\cdots ,d_k\\}$ be the set of distances between pairs of distinct points... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let \\[T=\\{(tq,1-t) \\in\\mathbb R^{2}| t \\in [0,1],q\\in\\mathbb Q\\}\\]Prove that each continuous function $f: T\\longrightarrow T$ has a fixed point.",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A set $\\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\\mathcal{S},$ the arithmetic mean of the set of values obtained by deleti... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $s(n)$ denote the sum of the digits (in base ten) of a positive integer $n$. Compute the number of positive integers $n$ at most $10^{4}$ that satisfy $$s(11 n)=2 s(n)$$",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "a) Find two sets $X,Y$ such that $X\\cap Y =\\emptyset$ , $X\\cup Y = \\mathbb Q^{\\star}_{+}$ and $Y = \\{a\\cdot b \\mid a,b \\in X \\}$ . \r\n\r\nb) Find two sets $U,V$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Esmeralda writes $2n$ real numbers $x_1, x_2, \\dots , x_{2n}$ , all belonging to the interval $[0, 1]$ , around a circle and multiplies all the pairs of numbers neighboring t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S_{7}$ denote all the permutations of $1,2, \\ldots, 7$. For any \\pi \\in S_{7}$, let $f(\\pi)$ be the smallest positive integer $i$ such that \\pi(1), \\pi(2), \\ldots, \\p... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "2.3 The bubble is initially formed with warmer air inside. Find the minimum numerical value of $T_{i}$ such that the bubble can float in still air. Use $T_{a}=300 \\mathrm{~K}, \\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The numbers $1, 2, \\ldots, 2012$ are written on a blackboard. Each minute, a student goes up to the board, chooses two numbers $x$ and $y$ , erases them, and writes the numb... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For any integer $k$, write $f_{k}(x)=\\left(1+x^{2}\\right)^{-1-k}$. When $k \\geq 1$, find constants $c_{1}, c_{2}$ such that the function $y=\\left(S f_{k}\\right)(x)$ solves a ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define the sequence $a_0,a_1,a_2,\\hdots$ by $a_n=2^n+2^{\\lfloor n/2\\rfloor}$ . Prove that there are infinitely many terms of the sequence which can be expressed as a sum of ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$ , define \\[g(n) = \\gcd\\left\\{0! n!, 1! (n-1)!, 2 (n-2)!, \\ldots, k!(n-k)!, \\ldots, n! 0!\\right\\}.\\] Find the sum of all $n \\leq 25$ for ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \\geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Denote $g(k)$ as the greatest odd divisor of $k$ . Put $f(k) = \\dfrac{k}{2} + \\dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1,... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $A B C$ be a solid right triangle $(A B=5 s, A C=12 s$, and $B C=13 s$ ) with uniform charge density $\\sigma$. Let $D$ be the midpoint of $B C$. We denote the electric potent... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider an optical system consisting of two thin lenses sharing the same optical axis. When a cuboid with a side parallel to the optical axis is placed to the left of the left le... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "On a sphere with radius $1$ , a point $ P $ is given. Three mutually perpendicular the rays emanating from the point $ P $ intersect the sphere at the points $ A $ , $ B $ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Point $B$ is in the exterior of the regular $n$ -sided polygon $A_1A_2\\cdots A_n$ , and $A_1A_2B$ is an equilateral triangle. What is the largest value of $n$ for which $A_1$ , $... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given are the points $A_0 = (0,0,0), A_1 = (1,0,0), A_2 = (0,1,0), A_3 = (0,0,1)$ in the space. Let $P_{ij} (i,j \\in 0,1,2,3)$ be the point determined by the equality: $\\ov... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $z = \\cos \\frac{4 \\pi}{7} + i \\sin \\frac{4 \\pi}{7}.$ Compute\n\\[\\frac{z}{1 + z^2} + \\frac{z^2}{1 + z^4} + \\frac{z^3}{1 + z^6}.\\]",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose you have a sphere tangent to the $xy$ -plane with its center having positive $z$ -coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$ -plane, it give... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Define the function $f(x) = \\lfloor x \\rfloor + \\lfloor \\sqrt{x} \\rfloor + \\lfloor \\sqrt{\\sqrt{x}} \\rfloor$ for all positive real numbers $x$ . How many integers from ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $b$ be a positive integer. Grogg writes down a sequence whose first term is $1$ . Each term after that is the total number of digits in all the previous terms of the seque... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For each integer $x$ with $1 \\leq x \\leq 10$, a point is randomly placed at either $(x, 1)$ or $(x,-1)$ with equal probability. What is the expected area of the convex hull of t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let triangle $ABC$ be a right triangle in the $xy$ -plane with a right angle at $C_{}$ . Given that the length of the hypotenuse $AB$ is $60$ , and that the medians through $A$ an... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the im... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S_{0}$ be a unit square in the Cartesian plane with horizontal and vertical sides. For any $n>0$, the shape $S_{n}$ is formed by adjoining 9 copies of $S_{n-1}$ in a $3 \\tim... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find any quadruple of positive integers $(a, b, c, d)$ satisfying $a^{3}+b^{4}+c^{5}=d^{11}$ and $a b c<10^{5}$.",
"role": "user"
},
{
"content": "<think>\nAlright, le... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many functions $f$ from \\{-1005, \\ldots, 1005\\} to \\{-2010, \\ldots, 2010\\} are there such that the following two conditions are satisfied? - If $a<b$ then $f(a)<f(b)$. -... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $n \\ge 2$ be an integer, and let $O$ be the $n \\times n$ matrix whose entries are all equal to $0$ . Two distinct entries of the matrix are chosen uniformly at random... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$ , and then written in nondecreasing order as $x_1\\le x_2\\le\\cdots\\le x_{2012}$ . I... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Dipole Conductor An (ideal) electric dipole of magnitude $p=1 \\times 10^{-6} \\mathrm{C} \\cdot \\mathrm{m}$ is placed at a distance $a=0.05 \\mathrm{~m}$ away from the center of... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For an arbitrary positive integer $m$ , not divisible by $3$ , consider the permutation $x \\mapsto 3x \\pmod{m}$ on the set $\\{ 1,2,\\dotsc ,m-1\\}$ . This permutation can ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Among all ellipses with center at the origin, exactly one such ellipse is tangent to the graph of the curve $x^3 - 6x^2y + 3xy^2 + y^3 + 9x^2 - 9xy + 9y^2 = 0$ at three distinct... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $p(x)$ be the monic cubic polynomial with roots $\\sin^2(1^{\\circ})$ , $\\sin^2(3^{\\circ})$ , and $\\sin^2(9^{\\circ})$ . Suppose that $p\\left(\\frac{1}{4}\\right)=\\... |
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