prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $k$ be the smallest positive integer with the following property: There are distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$ p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right) $$ has exactly $k$ nonzero coefficients. Find, with ... | 3 |
Alice drew a regular 2021-gon in the plane. Bob then labelled each vertex of the 2021-gon with a real number, in such a way that the labels of consecutive vertices differ by at most 1. Then, for every pair of non-consecutive vertices whose labels differ by at most 1, Alice drew a diagonal connecting them. Let $d$ be th... | 2018 |
We say a $2023$-tuple of nonnegative integers $\left(a_{1}, a_{2}, \ldots a_{2023}\right)$ is sweet if the following conditions hold: - $a_{1}+a_{2}+\ldots+a_{2023}=2023$, - $\frac{a_{1}}{2^{1}}+\frac{a_{2}}{2^{2}}+\ldots+\frac{a_{2023}}{2^{2023}} \leqslant 1$. Determine the greatest positive integer $L$ such that $$ a... | 22228 |
Every positive integer is marked with a number from the set $\{0,1,2\}$, according to the following rule: if a positive integer $k$ is marked with $j$, then the integer $k+j$ is marked with 0. Let $S$ denote the sum of marks of the first 2019 positive integers. Determine the maximum possible value of $S$. | 2021 |
For a positive integer $a$, define a sequence of integers $x_{1},x_{2},\ldots$ by letting $x_{1}=a$ and $x_{n+1}=2x_{n}+1$. Let $y_{n}=2^{x_{n}}-1$. Determine the largest possible $k$ such that, for some positive integer $a$, the numbers $y_{1},\ldots,y_{k}$ are all prime. | 2 |
A rectangle $ABCD$ is made up of five small congruent rectangles as shown in the given figure. Find the perimeter, in cm, of $ABCD$ if its area is $6750 cm^2$. | 330 |
Two squares of different sizes overlap as shown in the given figure. What is the difference between the non-overlapping areas? | 20 |
Evaluate $$\begin{aligned} &\begin{aligned}&~~~~\quad1\left(\frac11+\frac12+\frac13+\frac14+\frac15+\frac16+\frac17+\frac18+\frac19+\frac1{10}\right)\end{aligned} \\ &+\quad3\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}\right) \\ &+\quad5\left(\frac13... | 55 |
Color the surfaces of a cube of dimension $5\times 5\times 5$ red, and then cut the cube into smaller cubes of dimension $1\times 1\times 1$. Take out all the smaller cubes which have at least one red surface and fix a cuboid, keeping the surfaces of the cuboid red. Now what is the maximum possible volume of the cuboid... | 96 |
Given that $A^4=75600\times B$. If $A$ and $B$ are positive integers, find the smallest value of $B$. | 25725 |
A box contains $4000$ to $6000$ candies. When the candies are evenly distributed to $5$, $6$, $7$, $8$, or $9$ children, there is always one candy left. If the candies are packaged into small bags each having the same number of candies, what is the largest number of candies below $4000$ in each bag so that no candies a... | 71 |
There are positive integers $k, n, m$ such that $\frac{19}{20}<\frac{1}{k}+\frac{1}{n}+\frac{1}{m}<1$. What is the smallest possible value of $k+n+m$? | 12 |
The sum of the two three-digit integers, $6A2$ and $B34$, is divisible by $18$. What is the largest possible product of $A$ and $B$? | 36 |
Let $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}$, where $a$ and $b$ are different four-digit positive integers (natural numbers) and $c$ is a five-digit positive integer (natural number). What is the number $c$? | 12030 |
Let $x$ be a fraction between $\frac{35}{36}$ and $\frac{91}{183}$. If the denominator of $x$ is $455$ and the numerator and denominator have no common factor except $1$, how many possible values are there for $x$? | 136 |
$$\begin{aligned} 1+2 & =3 \\ 4+5+6 & =7+8 \\ 9+10+11+12 & =13+14+15\\ \vdots \end{aligned}$$ If this pattern is continued, find the last number in the $80$th row (e.g. the last number of the third row is $15$). | 6560 |
An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is $200$ and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle. | 56 |
For distinct positive integers $a$, $b < 2012$, define $f(a,b)$ to be the number of integers $k$ with $1 \le k < 2012$ such that the remainder when $ak$ divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a,b)$, where $a$ and $b$ range over all pairs of distinct positive... | 502 |
$P(x)$ is a polynomial of degree $3n$ such that $$\begin{array}{rcl}P(0)=P(3)=\cdots&=&P(3n)=2,\\P(1)=P(4)=\cdots&=&P(3n-2)=1,\\P(2)=P(5)=\cdots&=&P(3n-1)=0,\quad and\\&&P(3n+1)=730.\end{array}$$ Determine $n$. | 4 |
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of $a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$. | 1700 |
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer? Note. The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be $\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}$ | 337 |
A5. Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this r... | 290 |
Let $x_1,\ldots,x_{100}$ be nonnegative real numbers such that $x_{i} + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, .., 100$ (we put $x_{101}=x_{1}, x_{102}=x_{2}$). Find the maximal possible value of the sum $S=\sum\limits_{i=1}^{100}x_ix_{i+2}.$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide ... | 27 |
Evaluate the product $$ \prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64} $$ Hint: Telescoping product.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 79 |
Find the maximum value of $$ \int_{0}^{y} \sqrt{x^{4}+\left(y-y^{2}\right)^{2}} d x $$ for $0 \leq y \leq 1$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 4 |
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
Evaluate the product $$ \prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64} $$ The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 79 |
$\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}=$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
$\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}=\qquad$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
The pairs of numbers $(a, b)$ such that $\left|a+b t+t^{2}\right| \leq 1$ for $0 \leq t \leq 1$ fill a certain region in the $(a, b)$-plane. What is the area of this region?The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 29 |
What is the average straight line distance between two points on a sphere of radius 1?The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
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 answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n}$$ as a rational number.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
Let exp(t) denote e' and $F( x) = \frac {x^{* }}{\exp ( x^{3}) }\int _{0}^{x}\int _{0}^{x- u}\exp ( u^{3}+ v^{3})$ $dv$ $du.$ Find lim $_{x\to\infty}F(x)$ or prove that it does not exist.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
Evaluate $$ \lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n} $$ #*The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
Let $E(n)$ denote the largest integer $k$ such that $5^k$ is an integral divisor of the product $1^12^23^3\cdots n^n.$ Calculate $$\lim_{n\to\infty}\frac{E(n)}{n^{2}}$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 9 |
Evaluate the infinite product $$\prod_{n=2}^{\infty} \frac{n^{3}-1}{n^{3}+1}.$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
Let $C$ be a fixed unit circle in the Cartesian plane. For any convex polygon $P$ each of whose sides is tangent to $C$. let $N(P, h, k)$ be the number of points common to $P$ and the unit circle with center at ( $h, k$ ). Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the a... | 11 |
Let $\exp (t)$ denote $e^{\prime}$ and $$ F(x)=\frac{x^{4}}{\exp \left(x^{3}\right)} \int_{0}^{x} \int_{0}^{x-u} \exp \left(u^{3}+v^{3}\right) d v d u $$ Find $\lim _{x \rightarrow \infty} F(x)$ or prove that it does not exist.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n... | 11 |
For each continuous function $f:[0,1]\to\mathbb{R}$, let $I(f)=$ $\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \int _0^{1}x\left ( f( x) \right ) ^{2}$ $dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 17 |
Let $a_0=5/2$ and $a_k=a^2_{k-1}-2$ for $k\geq1.$ Compute $$\prod\limits_{k=0}^{\infty}\left(1-\frac{1}{a_k}\right)$$ in closed form.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 10 |
Determine the greatest possible value of $\sum_{i=1}^{10}\cos(3x_i)$ for real numbers $x_1,x_2,\ldots,x_{10}$ satisfying $\sum_{i=1}^{10}\cos(x_i)=0.$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 529 |
Suppose $X$ is a random variable that takes on only nonnegative integer values, with $E\left[X\right]=1,E\left[X^2\right]=2$,and $E\left[X^{3}\right]=5.$(Here $E\left[y\right]$denotes the expectation of the random variable $Y.)$ Determine the smallest possible value of the probability of the event $X=0.$The answer is i... | 4 |
Find the smallest constant C such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $$\int_{0}^{1} \left\vert P(x)\right\vert\, {\rm d}x\leq C\max _{x \in[0,1]} \left\vert P(x) \right\vert$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m... | 11 |
Determine the maximum value of the sum $$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$ over all sequences $a_{1},a_{2},a_{3},\cdots\$ of nonnegative real numbers satisfying $$\sum_{k=1}^{\infty}a_{k}=1.$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the val... | 5 |
Let $n$ be a positive integer, and let $V_n$ be the set of intger$\left(2n+1\right)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which$s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac1{q(\mathbf{v})}$... | 4041 |
The horizontal line $y=c$ intersects the curve $y=2 x-3 x^{3}$ in the first quadrant as in the figure. Find $c$ so that the areas of the two shaded regions are equal. The answer is in the form ... | 13 |
Evaluate$$\lim_{n\rightarrow\infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^{2}\left\{\frac{\pi}{2n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\}dx_{1}dx_{2}\cdots dx_{n}.$$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 3 |
Evaluate the infinite product $$\prod_{n=2}^{\infty}\frac{n^{3}-1}{n^{3}+1}$$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
Express $$\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{m^{2}n+mn^{2}+2mn}$$as a rational number.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 11 |
Evaluate the double series $$ \sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3 k-j-(k+n)^{2}} $$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 7 |
Let $A$ be the area of the region in the first quadrant bounded by the line $y=\frac{1}{2} x$, the $x$-axis, and the ellipse $\frac{1}{9} x^{2}+y^{2}=1$. Find the positive number $m$ such that $A$ is equal to the area of the region in the first quadrant bounded by the line $y=m x$, the $y$-axis, and the ellipse $\frac{... | 11 |
Let $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral $UWXY$ to the area of the triangle $PQR$. The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide t... | 13 |
In the figure, $ABCD$ is a square, $P$, $Q$, $R$, and $S$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ respectively. Find the ratio of the shaded area to the area of the square $ABCD$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 6 |
Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition: $a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$ Find the difference between the maximum value and the minimum value of the expression.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. P... | 1270 |
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provi... | 13 |
Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions. The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n. | 5 |
Determine the greatest possible value of $\sum_{i=1}^{10}\cos(3x_i)$ for real numbers $x_1,x_2,\ldots,x_{10}$ satisfying $\sum_{i=1}^{10}\cos(x_i)=0.$ The original answer is in \frac{m}{n} format, please give the value of m + n. | 529 |
Given real numbers $b_0,b_1,\ldots,b_{2019}$ with $b_{2019}\neq0$, let $z_1,z_2,\ldots,z_{2019}$ be the roots in the complex plane of the polynomial $$P(z)=\sum_{k=0}^{2019}b_kz^k.$$ Let $\mu=(|z_{1}|+\cdots+|z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots,z_{2019}$ to the origin. Determine the la... | 4037 |
For each real number $x$, let $$f(x)=\sum_{n\in S_x}\frac{1}{2^n},$$ where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x) ≥ L$ for all $x ∈ [0,1)?$ The original answer is in the format \frac{m}{n}, please provide the value of m + n. | 11 |
Let $$I(R)=\iint_{x^2+y^2\leq R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right)dxdy.$$ Find $$\lim_{R\to\infty}I(R),$$ or show that this limit does not exist. The original answer is in \frac{\sqrt{a}}{b}\pi\logc format, please provide the value of a+b+c. | 6 |
Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S. The original answer is in... | 8 |
Let $p$ be an odd prime number, and let ${\Bbb {F}}_p$ denote the field of integers modulo $p$. Let ${\Bbb {F}}_p[x]$ be the ring of polynomials over ${\Bbb {F}}_p$, and let $q(x)\in {\Bbb {F}}_p[x]$ be given by $$q(x)=\sum\limits_{k=1}^{p-1}a_kx^k,$$ where $$a_k=k^{(p-1)/2}\mod p.$$ Find the greatest nonnegative integ... | 1 |
Let A be the set of all integers $n$ such that $1\leq n\leq2021$ and $\gcd(n,2021)=1.$ For every nonnegative integer $j$, let $$S(j)=\sum_{n\in A}n^j.$$ Determine the smallest positive integer $k$ such that all values of $j$ not divisible by $k$ make $S(j)$ a multiple of 2021. Calculate the sum of all possible values o... | 88 |
Find the smallest constant C such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $$\int_{0}^{1} \left\vert P(x)\right\vert\, {\rm d}x\leq C\max _{x \in[0,1]} \left\vert P(x) \right\vert$$. The original answer is in \frac{m}{n} format, please give the value of m + n. | 11 |
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? The standard answer format is ... | 35 |
For a positive integer $N$, let $f_N^{1}$ be the function defined by $$f_N(x)=\sum_{n=0}^N\frac{N+1/2-n}{(N+1)(2n+1)}\sin((2n+1)x).$$ Determine the smallest constant $M$ such that $f_N(x)\leq M$ for all $N$ and all real $x.$ The original answer is in $\frac{m\pi}{n}$ format, please give the value of m + n. | 5 |
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form \frac{a\sqrt{b}- c}{d}, please provide the value of a + b + c + d. | 14 |
Let $f$ be a function on $[0,\infty)$, differentiable and satisfying $$f'(x)=-3f(x)+6f(2x)$$ for $x>0.$ Assume that $|f(x)|\leqslant e^{-\sqrt{x}}$ for $x\geqslant0$ (so that $f(x)$ tends rapidly to 0 as $x$ increases). For $n$ a non-negative integer, define $$\mu_n=\int_0^\infty x^nf(\:x\:)\:dx$$ (sometimes called the... | 5 |
Two real numbers $x$ and $y$ are chosen at random in the interval (0,1) with respect to the uniform distribution. What is the probability that the closest integer to $x/y$ is even? The original answer is in the form $r + s\pi$, please give the value of $r + s$. | 4 |
Let $p$ be an odd prime and let $\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are in the set $$ \left\{x^2: x \in \mathbf{Z}_p\right\} \cap\left\{y^2+1: y \in \mathbf{Z}_p\right\} ? $$ Please provide the number of elements in the intersection when p = 13. | 4 |
Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^xb^{1-x}\leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0<|u| ≤c$ and for all $x,0<x<1$. (Note: $\sinh u= (e^u- e^{-u})/2.$) Please provide the sum of the bounds of $u$ when $|u| = 1... | 2 |
Find all $\mathbf{v}\in\mathscr{P}$ such that the set $\mathscr{P}\setminus\{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathscr{P}$ can be partitioned into two sets of equal size and equal sum. The original answer is in ”vectors $(m, b)$ where $n \leq$ $b \leq j$ and $b$ is even. “format, Please prov... | 101 |
Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point (0,0) together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n.$ Determine, as a function of $n$, the numbe... | 51 |
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? The original... | 8 |
Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1\leq i\leq m$ and $1\leq j\leq n.$ There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1\leq i\leq m-1$ and $1\leq j\leq n-1.$ If a coin occupies the square $(i,j)$ with $i\leq m-1$ and $j\leq n-1$ and the squares $... | 0 |
Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2=2025$ and 2025-2016=9 is a perfect square. (Of the positive integers betwee... | 25 |
For all $n\geq1$, let $$a_n=\sum_{k=1}^{n-1}\frac{\sin\left(\frac{(2k-1)\pi}{2n}\right)}{\cos^2\left(\frac{(k-1)\pi}{2n}\right)\cos^2\left(\frac{k\pi}{2n}\right)}.$$ Determine $$\lim_{n\to\infty}\frac{a_n}{n^3}.$$ The original answer is in the format $\frac{m}{\pi^{n}}$. Please provide the value of m + n. | 11 |
Determine the maximum value of the sum $$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$ over all sequences $a_{1},a_{2},a_{3},\cdots$ of nonnegative real numbers satisfying $$\sum_{k=1}^{\infty}a_{k}=1.$$ The original answer is in \frac{m}{n} format, please give the value of m + n. | 5 |
Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S\subset\mathbb{R}^3$ with exactly $n$ elements such that $$S=\{v\times w:v,w\in S\}?$$ Please provide the sum of all possible values of $n$. | 8 |
Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1.$ Iteratively, for $n=0,1,2,\ldots$, a point $x_{n+1}$ is chosen uniformly from the interval $[0,x_n].$ Let Z be the smallest value of $n$ for which $x_n<\delta.$ Find the expected value of Z, as a function of $\delta$. Please provide the value when ... | 0 |
Let A be a $2n\times2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability 1/2. Find the expected value of $\det(A-A^{t})$ (as a function of n), where $A^{t}$ is the transpose of $A.$ The original answer is in \frac{(an)!}{b^{n}n!} format, please give the valu... | 6 |
Evaluate the sum $\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$ $=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$ (As u... | 4 |
Let $\mathscr{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq1,y\geq1$ and that satisfy the following two equations (where subscripts denote partial derivatives): $$\begin{array}{c}xf_x+yf_y=xy\ln(xy),\\x^2f_{xx}+y^2f_{yy}=xy.\end{array}$$ For each $f\in\mathscr{F}$,let $$m(f)... | 7 |
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over ... | 4041 |
A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a > b > c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers. Please provide... | 24 |
Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\leq y\leq x^{3}$,then$(f(x))^2\leq f(y)\leq(f(x))^3.$ Please provide the smallest value of c when assuming $f(x)=x^c$ and c is a specific constant value such that the function satisfies the ... | 1 |
For $0\leq p\leq1/2$,let $X_1,X_2,\ldots$ be independent random variables such that $$X_i=\begin{cases}1&with~ probability~p,\\-1&{with ~probability}~p,\\0&{with ~probability}~1-2p,\end{cases}$$ for all $i\geq1.$ Given a positive integer $n$ and integers $b,a_1,\ldots,a_n$, let $P(b,a_1,\ldots,a_n)$ denote the probabil... | 5 |
Determine which positive integers n have the following property: For all integers m that are relatively prime to n, there exists a permutation $\pi:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that $\pi(\pi(k))\equiv mk(\bmod n)$ for all $k\in\{1,2,\ldots,n\}$. Provide the sum of all such positive integers n. | 3 |
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k}).$ Evaluate $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}{k}.$$ The original answer is in the format $\frac{m\pi^j}{n}$, please provide the value of m + n + j. | 19 |
Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\ldots,n\},$ let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n.$ Let S be the n-by-n matrix whose $(i,j)$ entry is $s(i,j).$ For example, when $n=5$, we have $S=\begin{bmatrix}6&3&2&2&2\\3&0&1&0&1\\2&1&0&0&1\\2&0&0&0&1\\2&1&1... | 2 |
A $2 \times 3$ rectangle has vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The si... | 15 |
Evaluate \(\int_{0}^{a} \int_{0}^{b} e^{\max \left(b^{2} x^{2}, a^{2} y^{2}\right)} d y d x\), where \(a\) and \(b\) are positive. If \(a = 1\) and \(b = 1\), find the value of \(m + n\) in the expression \(\frac{e^{a^mb^n}-j}{ab}\), where the answer is in the format \(m + n + j\). | 5 |
A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form $\frac{a\sqrt{b}-c}{d}$, please give the value of a + b + c + d. | 14 |
Let f be a function on $[0, \infty)$, differentiable and satisfying $$ f^{\prime}(x)=-3 f(x)+6 f(2 x) $$ for $x>0$. Assume that $|f(x)| \leqslant e^{-\sqrt{x}}$ for $x \geqslant 0$ (so that $f(x)$ tends rapidly to 0 as x increases). For $n$ a non-negative integer, define $$ \mu_{n}=\int_{0}^{\infty} x^{n} f(x) d x $$ (... | 32 |
For any pair $(x, y)$ of real numbers, a sequence $\left(a_{n}(x, y)\right)_{n \geq 0}$ is defined as follows: $$\begin{aligned} a_{0}(x, y) & =x \\ a_{n+1}(x, y) & =\frac{\left(a_{n}(x, y)\right)^{2}+y^{2}}{2}, \quad \text { for all } n \geq 0. \end{aligned}$$ Find the area of the region $\left\{(x, y) \mid\left(a_{n}... | 5 |
Two real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x / y$ is even? The original answer is in the form $(m-\pi)/n$, please provide the value of m+n. | 9 |
Find all positive integer pairs $(x, y)$ such that $x^{2}-y^{119}=1$. Please provide the sum of all possible $x$ and $y$ values from the solutions.If the equation has no solution, then the answer is 0. | 0 |
$\frac{2020}{6063}$ | 8083 |
Evaluate $29 \frac{27}{28} \times 27\frac{14}{15}$. The original answer is in the form of $a + \frac{m}{n}$, where a is the integer part. Please provide the value of a + m + n. | 1358 |
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