prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Given positive integers $m$ and $n \geq m$, determine the largest number of dominoes ( $1 \times 2$ or $2 \times 1$ rectangles) that can be placed on a rectangular board with $m$ rows and $2 n$ columns consisting of cells ( $1 \times 1$ squares) so that: $\mathrm{(i)}$ each domino covers exactly two adjacent cells of t... | 27 |
Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight. The original answer is in the format $2^m$. Please provide the value of m + 1. | 1010 |
Let $n$ be an integer greater than 1, and let $X$ be a set of $n$ elements. If the union $A_{1} \cup \cdots \cup A_{k}$ is a proper subset of $X$, and no element of $X$ is in exactly one of $A_{i}$, then the set of non-empty subsets $A_{1}, \ldots, A_{k}$ of $X$ is compact. Find the maximum cardinality of the set of pr... | 198 |
Fix an integer $n \geq 2$. An $n \times n$ sieve is an $n \times n$ array with $n$ cells removed so that exactly one cell is removed from each row and column. For any positive integer $k$, a stick is a $1 \times k$ or $k \times 1$ array. For any sieve $A$, let $m(A)$ denote the minimum number of sticks required to part... | 198 |
Let $\mathbb{N}$ denote the set of all positive integers. Find a specific function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for $x = 1$ and $y = 1$, the expression $x^{2}-y^{2}+2 y(f(x)+f(y))$ becomes a square of an integer. Provide the value of $f(8)$. | 8 |
Find all pairs of positive real numbers $(x, y)$ such that $x y$ is an integer and $$ x+y=\left\lfloor x^{2}-y^{2}\right\rfloor. $$ Calculate the value of $x$ when $y=100$. | 101 |
Find all positive integers $a, b, n$ and prime numbers $p$ that satisfy the following conditions $$ a^{2013}+b^{2013}=p^{n} $$ Calculate the value of $n$ when $a=2$. | 2014 |
Olja writes down $n$ positive integers $a_{1}, a_{2}, \ldots, a_{n}$ smaller than $p_{n}$ where $p_{n}$ denotes the $n$-th prime number. Oleg can choose two (not necessarily different) numbers $x$ and $y$ and replace one of them with their product $x y$. If there are two equal numbers Oleg wins. Can Oleg guarantee a wi... | 2 |
Let $n$ be a positive integer. Let $B_{n}$ be the set of all binary strings of length $n$. For a binary string $s_{1} s_{2} \ldots s_{n}$, we define its twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_{b}$ with $1-s_{b}$. A string ... | 4 |
We are given a combination lock consisting of 6 rotating discs. Each disc consists of digits $0,1,2, \ldots, 9$, in that order (after digit 9 comes 0 ). Lock is opened by exactly one combination. A move consists of turning one of the discs one digit in any direction and the lock opens instantly if the current combinati... | 999998 |
We call a sequence of $n$ digits one or zero a code. Subsequence of a code is a palindrome if it is the same after we reverse the order of its digits. A palindrome is called nice if its digits occur consecutively in the code. (Code (1101) contains 10 palindromes, of which 6 are nice.) What is the least number of palind... | 4 |
Find all pairs $(x, y)$ of integers that satisfy the equation $$ x^{2} y+y^{2}=x^{3}. $$ Please provide the value of x + y for the pair that satisfies the equation. | -12 |
A friendly football match lasts 90 minutes. In this problem, we consider one of the teams, coached by Sir Alex, which plays with 11 players at all times.\na) Sir Alex wants for each of his players to play the same integer number of minutes, but each player has to play less than 60 minutes in total. What is the minimum ... | 35 |
Find all polynomials $P$ with integer coefficients such that $P(0) \neq 0$ and $$ P^{n}(m) \cdot P^{m}(n) $$ is a square of an integer for all nonnegative integers $n, m$. Remark : For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n\text{ if }k = 0$ and $P^k(n)=P(P^{k-1}(n))$ i... | 2 |
A positive integer $k \geqslant 3$ is called fibby if there exists a positive integer $n$ and positive integers $d_{1}<d_{2}<\ldots<d_{k}$ with the following properties: - $d_{j+2}=d_{j+1}+d_{j}$ for every $j$ satisfying $1 \leqslant j \leqslant k-2$, - $d_{1}, d_{2}, \ldots, d_{k}$ are divisors of $n$, - any other div... | 4 |
For all positive integers $n, k$, let $f(n, 2 k)$ be the number of ways an $n \times 2 k$ board can be fully covered by $n k$ dominoes of size $2 \times 1$. (For example, $f(2,2)=2$ and $f(3,2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2 k)$ is odd. Please provide th... | 0 |
The numbers $p$ and $q$ are prime and satisfy $$\frac{p}{p+1}+\frac{q+1}{q}=\frac{2 n}{n+2}$$ for some positive integer $n$. Please provide the sum of all possible values of $q-p$. | 10 |
Let $k$ be a positive integer. Lexi has a dictionary $\mathcal{D}$ consisting of some $k$-letter strings containing only the letters $A$ and $B$. Lexi would like to write either the letter $A$ or the letter $B$ in each cell of a $k \times k$ grid so that each column contains a string from $\mathcal{D}$ when read from t... | 2 |
For a sequence $a_{1}<a_{2}<\cdots<a_{n}$ of integers, a pair ( $a_{i}, a_{j}$ ) with $1 \leq i<$ $j \leq n$ is called \underline{interesting} if there exists a pair ( $a_{k}, a_{l}$ ) of integers with $1 \leq k<l \leq n$ such that $$\frac{a_{l}-a_{k}}{a_{j}-a_{i}}=2$$ For each $n \geq 3$, find the largest possible num... | 7 |
Find the largest positive integer \(d\) for which there exists a degree $d$ polynomial $P$ with real coefficients such that there are at most $d$ different values among $P(0), P(1), \ldots, P\left(d^{2}-\right.$ $d)$. | 3 |
Determine all integers $m$ for which the $m \times m$ square can be dissected into five rectangles, the side lengths of which are the integers $1,2,3, \ldots, 10$ in some order. Please provide the sum of all possible values of $m$. | 24 |
Determine all real constants $t$ such that whenever $a, b, c$ are the lengths of the sides of a triangle, then so are $a^{2}+b c t, b^{2}+c a t, c^{2}+a b t$.If the length of the interval in which \(t\) lies is \(\frac{a}{b}\), find \(a + b\). | 7 |
A domino is a $2 \times 1$ or $1 \times 2$ tile. Determine in how many ways exactly $n^{2}$ dominoes can be placed without overlapping on a $2 n \times 2 n$ chessboard so that every $2 \times 2$ square contains at least two uncovered unit squares which lie in the same row or column. Please provide the value when n = 1. | 4 |
Let $m$ be a positive integer. Consider a $4 m \times 4 m$ array of square unit cells. Two different cells are related to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are coloured blue, such that every cell is related to at least two blue cells. Determine... | 6 |
The $n$ contestants of an EGMO are named $C_{1}, \ldots, C_{n}$. After the competition they queue in front of the restaurant according to the following rules. - The Jury chooses the initial order of the contestants in the queue. - Every minute, the Jury chooses an integer $i$ with $1 \leq i \leq n$. - If contestant $C_... | 502 |
Determine whether for every real number $t$ such that $0<t<\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that $$ |x-m y|>t y $$ for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ).The original answer is in the form of $t<\left\{\frac{s_{n-1}}{s_... | 3 |
Let $n$ be a positive integer. Dominoes are placed on a $2 n \times 2 n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way. (A domino is a tile of size $2 \times 1$ or $1 \times 2$. ... | 21 |
Let $n$ be a positive integer and fix $2 n$ distinct points on a circumference. Split these points into $n$ pairs and join the points in each pair by an arrow (i.e., an oriented line segment). The resulting configuration is good if no two arrows cross, and there are no arrows $\overrightarrow{A B}$ and $\overrightarrow... | 924 |
Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide th... | 9 |
Let $A$ be a real $4 \times 2$ matrix and $B$ be a real $2 \times 4$ matrix such that $$ AB=\left(\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{array}\right) $$ Find $B A.$ Provide the sum of all elements in the resulting matrix. | 4 |
How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible numbers of nonzero coefficients. | 3 |
For each positive integer $k$, find the smallest number $n_{k}$ for which there exist real $n_{k} \times n_{k}$ matrices $A_{1}, A_{2}, \ldots, A_{k}$ such that all of the following conditions hold: (1) $A_{1}^{2}=A_{2}^{2}=\ldots=A_{k}^{2}=0$, (2) $A_{i} A_{j}=A_{j} A_{i}$ for all $1 \leq i, j \leq k$, and (3) $A_{1} ... | 12 |
Compute the sum of the series \( \sum_{k=0}^{\infty} \frac{1}{(4 k+1)(4 k+2)(4 k+3)(4 k+4)}=\frac{1}{1 \cdot 2 \cdot 3 \cdot 4}+\frac{1}{5 \cdot 6 \cdot 7 \cdot 8}+\cdots \). The original answer is in a special format：$\frac{\ln m}n-\frac\pi{p}.$ provide the sum of the denominators in the expression. | 30 |
(a) A sequence $x_{1}, x_{2}, \ldots$ of real numbers satisfies $$ x_{n+1}=x_{n} \cos x_{n} \quad \text { for all } \quad n \geq 1 $$ Determine if this sequence converges for all initial values $x_{1}$. (b) A sequence $y_{1}, y_{2}, \ldots$ of real numbers satisfies $$ y_{n+1}=y_{n} \sin y_{n} \quad \text { for all } \... | 1 |
Given an integer $n>1$, let $S_{n}$ be the group of permutations of the numbers $1,2, \\ldots, n$. Two players, A and B, play the following game. Taking turns, they select elements (one element at a time) from the group $S_{n}$. It is forbidden to select an element that has already been selected. The game ends when the... | 5 |
Consider the following sequence \(\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)\). Find all pairs \((\alpha, \beta)\) of positive real numbers such that \(\lim \limits_{n \rightarrow \infty} \frac{\sum\limits_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta\). The original answer is in \(\frac{\sqrt{m}}... | 5 |
We say that a subset of $\mathbb{R}^{n}$ is $k$-almost contained by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-generic if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $k$ and $n$, fin... | 8 |
Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum\limits_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number. Please provide the integer value of the sum. | 1 |
An $n \times n$ complex matrix $A$ is called $t$-normal if $A A^{t}=A^{t} A$ where $A^{t}$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Please provide the value of the dimension when n = 11. The answer should b... | 134 |
Determine all complex numbers $\lambda$ for which there exist a positive integer $n$ and a real $n \times n$ matrix $A$ such that $A^{2}=A^{T}$ and $\lambda$ is an eigenvalue of $A$. The answer is $\frac{m \pm \sqrt{n} i}{p}$ ,please give the sum of m,n and p. | 4 |
Determine all rational numbers $a$ for which the matrix $$ \left(\begin{array}{cccc} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a \\ 0 & 1 & a & -a \end{array}\right) $$ is the square of a matrix with all rational entries. Provide the integer value of $a$ that satisfies the condition. | 0 |
For $R>1$ let $\mathcal{D}_{R}=\left\{(a, b) \in \mathbb{Z}^{2}: 0<a^{2}+b^{2}<R\right\}$. Compute $$ \lim _{R \rightarrow \infty} \sum_{(a, b) \in \mathcal{D}_{R}} \frac{(-1)^{a+b}}{a^{2}+b^{2}} $$ The original answer is in the format $m\pi \log n$. Please provide the integer value of m and n. | 1 |
Let $\Omega=\left\{(x, y, z) \in \mathbb{Z}^{3}: y+1 \geq x \geq y \geq z \geq 0\right\}$. A frog moves along the points of $\Omega$ by jumps of length 1. For every positive integer $n$, determine the number of paths the frog can take to reach $(n, n, n)$ starting from $(0,0,0)$ in exactly $3 n$ jumps. The original ans... | 843 |
Evaluate the product $$ \prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64} $$ The original answer is in \(\frac{m}{n}\) format, please give the value of m + n. | 79 |
A four-digit number $Y E A R$ is called very good if the system $$ \begin{aligned} & Y x+E y+A z+R w=Y \\ & R x+Y y+E z+A w=E \\ & A x+R y+Y z+E w=A \\ & E x+A y+R z+Y w=R \end{aligned} $$ of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good YEARs in the 21st century and... | 12309 |
Find all twice continuously differentiable functions $f: \mathbb{R} \rightarrow(0,+\infty)$ satisfying $$ f^{\prime \prime}(x) f(x) \geq 2\left(f^{\prime}(x)\right)^{2} $$ for all $x \in \mathbb{R}$. Please provide the sum of the solutions $f^{\prime}$ and $f^{\prime \prime}$. | 0 |
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of all possible remainders. | 5 |
Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying $$ |a|=|b|=1 \quad \text { and } \quad a+b+a \bar{b} \in \mathbb{R}. $$ (proposed by Mike Daas, Universiteit Leiden) Please provide the sum of all possible values of a and b. | 0 |
Find a necessary and sufficient condition on the positive integer $n$ that the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has a rational root. Please provide the integer value of $n$ that satisfies this condition. | 1 |
For $n=1,2, \ldots$ let $$ S_{n}=\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)-\log (\sqrt{n}) $$ where $\log$ denotes the natural logarithm. Find $\lim\limits _{n \rightarrow \infty} S_{n}$. The original answer is in $-\frac{m}{n}$ format, please give the value of m + n. | 5 |
We throw a dice (which selects one of the numbers $1,2, \ldots, 6$ with equal probability) $n$ times. What is the probability that the sum of the values is divisible by 5? What is the value of the probability when n = 1/6? | 0 |
Let $n$ be a positive integer. Consider an $n \times n$ matrix with entries $1,2, \ldots, n^{2}$ written in order starting top left and moving along each row in turn left-to-right. We choose $n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the ... | 5 |
Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area. Please provide the sum of the coordinates of the points, when a = 2. | 8 |
A penny in a corner. A circle moves so that it is continually in contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle. Please provide the num of the possible condition that $x_{i}^{2} \leq r^{2}$. | 3 |
Find a necessary and sufficient condition on the positive integer $\boldsymbol{n}$ that the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has a rational root. Please provide the integer value of $n$ that satisfies this condition. | 1 |
Assuming that the roots of $x^{3}+p x^{2}+q x+r=0$ are all real and positive, find the relation between $p, q$, and $r$ which is a necessary and sufficient condition that the roots may be the cosines of the angles of a triangle. Given that the original answer is in the form of an equation $p^n-mq-jr=1$, provide the sum... | 6 |
The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the... | 10 |
Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$. | 9 |
Find the length of the curve $y^{2}=x^{3}$ from the origin to the point where the tangent makes an angle of $45^{\circ}$ with the $x$-axis. The original answer involves an integral that evaluates to a form with square roots. The original answer form is $\ frac {a} {b} (c \ sqrt {d}-e ）Find the value of a+b+c+d+e. | 40 |
What is the maximum of $\left|z^{3}-z+2\right|$, where $z$ is a complex number with $|z|=1$? The original answer is in the format \sqrt{m}, please provide the value of m. | 13 |
Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$ The original answer is in the format $\frac{1}{2} \sqrt{3} abc$, Let abc=2 \ sqrt {3}, find the value of V. | 3 |
Find all solutions of $n^{m}=m^{n}$ in integers $n$ and $m(n\neq m)$. Prove that you have obtained all of them. Please provide the sum of all values of $m + n$ for each solution. | 0 |
(i) A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. Let a=2，The original answer is in the format $m \pi$, Find the value of m. | 8 |
The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down. | 5 |
Evaluate in closed form $$\sum_{k=1}^{n}\binom{n}{k} k^{2}$$ Note: $$\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k}$$Please provide the value when \(n = 3\). | 24 |
Evaluate $$\lim _{n \rightarrow \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^{2}+j^{2}}$$ The original answer is in \frac{m\pi}{n} format. Please provide the value of m + n. | 3 |
A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per seco... | 5 |
Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. The original answer... | 27 |
If $x_{1}, \ldots, x_{n}$ denote the $n$th roots of unity, evaluate $$ \pi\left(x_{i}-x_{j}\right)^{2} \quad(i<j) $$What is the value when n=3。 | -27 |
A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n\to\infty} p(n)$. The original answer is in \frac{m}{n} format, please give the value... | 9 |
In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A +... | 23 |
For a fixed positive integer $n$ let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $0 \leq x_{k} \leq 1$ for $k=1,2, \ldots, n$. Determine the maximum value, as a function of $n$, of the sum of the $n(n-1) / 2$ terms: $$ \sum_{\substack{i, j=1 \\ i<j}}^{n}\left|x_{i}-x_{j}\right|. $$ Please provide the value... | 2 |
Determine all rational values for which $a, b, c$ are the roots of $$x^{3}+a x^{2}+b x+c=0$$ How many pairs of \(a\), \(b\), \(c\) satisfy the given conditions? | 3 |
Find the volume of the four-dimensional hypersphere $x^{2}+y^{2}+z^{2}+t^{2}=r^{2}$, and also the hypervolume of its interior $x^{2}+y^{2}+z^{2}+t^{2}<r^{2}$. If \(r = 2\), given that the sum of these two parts can be written in the form of \(k\pi^{2}\), find the value of \(k\). | 24 |
Solve the system of differential equations$$\begin{aligned}& \frac{d x}{d t}=x+y-3 \\& \frac{d y}{d t}=-2 x+3 y+1\end{aligned}$$subject to the conditions $x=y=0$ for $t=0$.When \(t = \pi\), the value of \(y\) can be written in the form of \(a + be^{c\pi}\). Find the value of \(a + b + c\). | 2 |
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elemen... | 8 |
For each nonnegative integer $k$, let $d(k)$ denote the number of I's in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)-2$). Let $m$ be a positive integer. Express $$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$ in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.Let \(m = 2\... | 5 |
Evaluate $$\lim_{n \to \infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^2 \{\frac{\pi}{2n}(x_{1}+x_{2}+\cdots x_{n})\}dx_{1}dx_{2}\cdots dx_{n}$$. The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\). | 3 |
Evaluate $$\int_{0}^{\infty}\frac{\arctan(\pi x)-\arctan x}{x}dx$$. The original answer is in the form of $(frac{m\pi}{n}})\ln\pi$. Please provide the value of $m + n$. | 3 |
Let $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$. The original answer is in the... | 15 |
Assume that the differential equation $$ y^{\prime \prime \prime}+p(x) y^{\prime \prime}+q(x) y^{\prime}+r(x) y=0 $$ has solutions $y_{1}(x), y_{2}(x)$, and $y_{3}(x)$ on the whole real line such that $$ y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1 $$ for all real $x$. Let $$ f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y... | 10 |
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r + s\sqrt{t}$ with $r$, $s$, and $t$ as positive integers. Please provide the value of $r + s + t$. | 27 |
A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one ... | 19 |
In how many ways can the integers from 1 to n be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it? Please provide the number of ways for $n = 6$. | 32 |
Let 0<a<b. Evaluate $$\lim_{t \to 0} \left \{ {\int_{0}^{1} [bx+a(1-x)]^{t}dx}\right \} ^{1/t}$$[The final answer should not involve any operations other than addition, subtraction, multiplication, division, and exponentiation.] When \(a = 1\) and \(b = 2\), the limit can be written as \(\frac{a}{be}\), where both \(a\... | 5 |
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$. and let $M_{n}$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$\left.m_{ij}=\left\{\begin{array}{llll}x&\text{if}\quad i=j,\\a&\text{if}\quad i\neq j&\text{and}&i+j\text{is even},\\b&\text{if}\quad i\neq j&\text{an... | 27 |
Let $A(x, y)$ denote the number of points $(m, n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^{2}+n^{2} \leqslant x^{2}+y^{2}$. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. Express $$ \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y $$ as a polynomial in $g$. The original... | 1 |
Let $P$ be a non-self-intersecting closed polygon with $n$ sides. Let its vertices be $P_1$, $P_{2},\cdots,P_{n}.$ Iet $m$ other points$,Q_1,Q_{2},\cdots,Q_{m}$ interior to $P$ be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1,\cdots,Q_m$ are connected by line segments s... | 11 |
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$. | 5 |
Express the sum \( \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n} \) as a rational number. The original answer is in the form of \( \frac{m}{n} \). Please provide the value of \( m + n \). | 11 |
Evaluate the following limit: \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}. The original answer is in the format $\frac{m}{n}$, please give the value of $m + n$. | 3 |
Let R be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x + y + z\leqslant 1$. Let $w=1-x-y-z$. Express the value of the triple integral $$\iiint\limits_{R}x^{1}y^{9}z^{8}w^{4}dx dy dz$$ in the form $\frac{a!b!c!d!}{n!}$, where a, b, c, d, and n are positive integers. Please prov... | 47 |
Let $A(x, y)$ denote the number of points ($m, n$) with integer coordinates $m$ and $n$ that satisfy $m^{2}+n^{2} \leqslant x^{2}+y^{2}$ in the plane. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. If $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y=\pi(ag-b)^{c}$$ Find the value of $a+b+c$. | 5 |
A game of solitaire is played as follows. After each play, according to the outcome, the player receives either a or b points (a and b are positive integers with a greater than b), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is ... | 19 |
For which real numbers $a$, the sequence defined by the initial condition $u_{0}=a$ and the recursion $u_{n+1}=2u_{n}-n^{2}$ has $u_{n}>0$ for all $n>0$? Give the minimum value of $a$. | 3 |
Suppose A is a three-dimensional $a × b × c$ rectangular brick with $a, b, c > 0$. Suppose B is the set of all points whose distance from a point in A is at most 1 (in particular, B contains A), and the volume of B is $a b c+m(a b+a c+b c)+\pi(a+b+c)+\frac{n \pi}{p}$. Find the value of $m+n+p$. | 9 |
Let $V$ be the region in the Cartesian plane consisting of all points ( x , y ) satisfying the simultaneous conditions $$\left\vert x \right\vert\leqslant y\leqslant\left\vert x \right\vert+3 \quad and \quad y\leqslant4$$ Find the centroid ( $\overline{x}$ , $\overline{y}$ ) of $V$. The original answer for $\overline{y... | 18 |
Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Please provide the value of a-b. Here $[x]$ is defined to be the integer such that $[x] \leqq x<[x]+1$ a... | 3 |
For a positive integer $n$, let $f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)$. Find the smallest $n$ such that $\left|f_n^{\prime \prime}(0)\right|>2023$. Please provide the value of $n$. | 18 |
An unbiased coin is tossed $n$ times. $H$ is the number of heads and $T$ is the number of tails. What is the expected value of $|H-T|$? In other words, evaluate in closed form: $$ \frac{1}{2^{n-1}} \sum_{k<n / 2}(n-2 k)\binom{n}{k} $$ (In this problem, "closed form" means a form that does not involve a series. The give... | 5 |
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