messages listlengths 3 3 |
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[
{
"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": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let $\\f... |
[
{
"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)=\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a \\( 2018 \\times 2019 \\) board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose som... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "a. What is the electric potential at a corner of the same cube? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $a,$ $\\sigma(a)$ denotes the sum of the positive integer divisors of $a$ . Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n \\geq 2$ be a natural. Define $$ X = \\{ (a_1,a_2,\\cdots,a_n) | a_k \\in \\{0,1,2,\\cdots,k\\}, k = 1,2,\\cdots,n \\} $$ .\nFor any two elements $s = (s_1,s_2,\\cdo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle{PQR}$ be a right triangle with $PQ = 90$ , $PR = 120$ , and $QR = 150$ . Let $C_{1}$ be the inscribed circle. Construct $\\overline{ST}$ with $S$ on $\\overline{P... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose X is N(0,1), \\epsilon is N(0, \\sigma^2) and is independent of X, and Y = X + \\epsilon. A statistician observes the value of Y and must decide whether the (unobserved) ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ be a positive integer. An $n \\times n$ matrix (a rectangular array of numbers with $n$ rows and $n$ columns) is said to be a platinum matrix if:\n[list=i]\n[*] t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguish... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "When the vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ -1 \\\\ 2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\\ 3 \\\\ 0 \\end{pmatrix}$ are both projected onto the ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a parlor game, the magician asks one of the participants to think of a three digit number $(abc)$ where $a$ , $b$ , and $c$ represent digits in base $10$ in the order indicate... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$ .",
"role... |
[
{
"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) Fo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A4 Define $x_{\\mathrm{g}}=h v_{\\mathrm{g}} / k_{\\mathrm{B}} T_{\\mathrm{s}}$ where $E_{\\mathrm{g}}=h v_{\\mathrm{g}}$. Express the useful output power of the cell, $P_{\\text... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $d \\ge 2$ be an integer. Prove that there exists a constant $C(d)$ such that the following holds: For any convex polytope $K\\subset \\mathbb{R}^d$ , which is symmetric... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In general, whenever an electric and a magnetic field are at an angle to each other, energy is transferred; for example, this principle is the reason electromagnetic radiation tr... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C=\\frac{5}{9}(F-32)$ . An integer Fahrenheit temperature is conver... |
[
{
"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 ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers less than or equal to $2017$ whose base-three representation contains no digit equal to $0$ .",
"role": "user"
},
{
"content": "<... |
[
{
"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 $x_1\\le x_2\\le\\cdots\\le x_{2012}$ . ... |
[
{
"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": "Let $\\rho:G\\to GL(V)$ be a representation of a finite $p$ -group $G$ over a field of characteristic $p$ . Prove that if the restriction of the linear map $\\sum_{g\\in G... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a)\tDecide whether there exist two decimal digits $a$ and $b$ , such that every integer with decimal representation $ab222 ... 231$ is divisible by $73$ .\n(b)\tDecide whe... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given integer $m\\geq2$ , $x_1,...,x_m$ are non-negative real numbers,prove that: $$ (m-1)^{m-1}(x_1^m+...+x_m^m)\\geq(x_1+...+x_m)^m-m^mx_1...x_m $$ and please find out when... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let the number $x$ . Using multiply and division operations of any 2 given or already given numbers we can obtain powers with natural exponent of the number $x$ (for example,... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The diameter of the planet Tatooine is 10465 km. If 100 million 1-GeV neutrinos pass through Tatooine, how many will interact?\n\nData\n\n\\sigma=0.7\\times10^{-38}cm^{2}/nucleon... |
[
{
"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": "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 $\\ove... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We define a binary operation $\\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \\star B$ is the third vertex of the equilateral triangle ABC ori... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$ , and $DE=12$ . Denote $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$ .",
... |
[
{
"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 th... |
[
{
"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": "(a) Let $n$ is a positive integer. Calculate $\\displaystyle \\int_0^1 x^{n-1}\\ln x\\,dx$ .\n\n(b) Calculate $\\displaystyle \\sum_{n=0}^{\\infty}(-1)^n\\left(\\frac{1}{(n+1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z=a+bi$ be the complex number with $\\vert z \\vert = 5$ and $b > 0$ such that the distance between $(1+2i)z^3$ and $z^5$ is maximized, and let $z^4 = c+di$ . Find $c+d$ .",... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbb{N}_0$ and $\\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\\mathbb{N}_0\\rightarrow\\mathbb{Z}$ be a f... |
[
{
"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": " $K(x, y), f(x)$ and $g(x)$ are positive and continuous for $x, y \\subseteq [0, 1]$ . $\\int_{0}^{1} f(y) K(x, y) dy = g(x)$ and $\\int_{0}^{1} g(y) K(x, y) dy = f(x)$ f... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\overline{CH}$ be an altitude of $\\triangle ABC$ . Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH^{}_{}$ are tangent to... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the fission of $^{236}U$ into two equal mass prompt fission fragments and zero free neutrons. What is the value of proton number Z giving the most stable nucleus with th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Fibonacci numbers $F_1, F_2, F_3, \\ldots$ are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for each integer $n \\ge 1$ . Let $P$ be the unique polynomia... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r... |
[
{
"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": "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.",
"role": "user"
},
{
"content": "<think>\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC,$ $AB = 13,$ $BC = 14,$ $AC = 15,$ and point $G$ is the intersection of the medians. Points $A',$ $B',$ and $C',$ are the images of $A,$ $B,$ and $C,$ respective... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An ideal gas is expanded from $\\left(\\mathrm{p}_{1}, \\mathrm{~V}_{1}, \\mathrm{~T}_{1}\\right)$ to $\\left(\\mathrm{p}_{2}, \\mathrm{~V}_{2}, \\mathrm{~T}_{2}\\right)$ under d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the exact value of\n\\[\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{3 \\pi}{7} \\right)... |
[
{
"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": "Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A spy is located on a one-dimensional line. At time 0, the spy is at location A. With each time interval, the spy moves B units to the right (if B is negative, the spy is moving ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For real constant numbers $ a,\\ b,\\ c,\\ d,$ consider the function $ f(x) \\equal{} ax^3 \\plus{} bx^2 \\plus{} cx \\plus{} d$ such that $ f( \\minus{} 1) \\equal{} 0,\\ f... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all real $x \\in [0, 2 \\pi]$ such that\n\\[\\tan 7x - \\sin 6x = \\cos 4x - \\cot 7x.\\]Enter all the real solutions, separated by commas.",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive value of $x,$ in degrees, for which\n\\[\\tan (x + 100^{\\circ}) = \\tan (x + 50^{\\circ}) \\tan x \\tan (x - 50^{\\circ}).\\]",
"role": "user... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lambda$ be a constant, $0 \\le \\lambda \\le 4,$ and let $f : [0,1] \\to [0,1]$ be defined by\n\\[f(x) = \\lambda x(1 - x).\\]Find the values of $\\lambda,$ $0 \\le \\lamb... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a linear harmonic oscillator and let, Ψ0 and Ψ 1 be its real, normalized ground and first excited state energy eigenfunctions respectively.\nLet Ψ = AΨ0 + BΨ1 with A and... |
[
{
"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 $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all twice differentiable functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that $$ f''(x) \\cos(f(x))\\geq(f'(x))^2 \\sin(f(x)) $$ for every $x\\in \\mathbb{R}$ .\n\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$",
"role": "user"
},
{
"content": "<think>\nAlright, so... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the additive group $\\mathbb{Z}^{2}$ . Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$ .\nLet $H_{xy}$ be the smallest subgroup containin... |
[
{
"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 \\ti... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For positive integers $i = 2, 3, \\ldots, 2020$ , let \\[\na_i = \\frac{\\sqrt{3i^2+2i-1}}{i^3-i}.\n\\]Let $x_2$ , $\\ldots$ , $x_{2020}$ be positive reals such that $x_2^4... |
[
{
"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": "Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all po... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the last few decades, reverberation mapping, interferometry and various other observations across the electromagnetic spectrum, have been used to probe the inner structure of ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of $500$ points. Alpha scored $16... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\theta$ be a constant number such that $0\\leq \\theta \\leq \\pi.$ \r\nEvaluate \r\n\r\n\\[ \\int_0^{2\\pi} \\sin 8x|\\sin (x-\\theta)|\\ dx. \\]",
"role": "user"
... |
[
{
"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": "Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\\cdots,P_{n},$ among $ n$ points no three points are ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\\in G$ we denote $H^x = \\{ xhx^{-1} \\mid h \\in H \\}$ ... |
[
{
"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": "Let \\[P(x) = (2x^4 - 26x^3 + ax^2 + bx + c)(5x^4 - 80x^3 + dx^2 + ex + f),\\]where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A train of length $100 \\mathrm{~m}$ and mass $10^{5} \\mathrm{~kg}$ is travelling at $20 \\mathrm{~m} / \\mathrm{s}$ along a straight track. The driver engages the brakes and th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces label... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $a,\\,b,$ and $c$ are three complex numbers with product $1$ . Assume that none of $a,\\,b,$ and $c$ are real or have absolute value $1$ . Define\n\\begin{tabul... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "vi. Eventually we cannot neglect the effect of the pressure on the dynamics of the gas and the collapse stops at $r=r_{4}$ (with $r_{4} \\ll r_{3}$ ). However, the radiation loss... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An IPv4 packet contains the following data (in hexadecimal value) in the IP header: 4500 0034 B612 4000 4006 6F80 0A00 008B 5BC6 AEE0 . Does the header contains error?",
"rol... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $ a_1, a_2, \\ldots, a_{2n}$ be $ 2n$ real numbers satisfying the condition $ \\sum_{i \\equal{} 1}^{2n \\minus{} 1} (a_{i \\plus{} 1} \\minus{} a_i)^2 \\equal{} 1$ . F... |
[
{
"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 $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$ . 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": "A cube of mass $m$ slides down the felt end of a ramp semicircular of radius $h$ , reaching a height $h/2$ at the opposite extreme.\nFind the numerical coefficient of fricti... |
[
{
"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": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\\tau (1)=1$ and $\\tau(6) =4.$ Define $S(n)$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": " $(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the ce... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$ .\nJulia erases one of the numbers. When Julia multiplies the remaining $199$ numbe... |
[
{
"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": "<thi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$ \\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}. $$ ",
"role": "user"
},
{... |
[
{
"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": "Three spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respective... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The terms of the sequence $\\{a_i\\}$ defined by $a_{n + 2} = \\frac {a_n + 2009} {1 + a_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $a_1 +... |
[
{
"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": "For each integer $n\\geq3$ , let $f(n)$ be the number of $3$ -element subsets of the vertices of a regular $n$ -gon that are the vertices of an isosceles triangle (including equi... |
[
{
"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": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the largest positive integer $k{}$ for which there exists a convex polyhedron $\\mathcal{P}$ with 2022 edges, which satisfies the following properties:\n[list]\n[*]The degre... |
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