Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial. [i]Ashwin Sah[/i]

Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

A positive integer $n$ between $1$ and $N=2007^{2007}$ inclusive is selected at random. If $a$ and $b$ are natural numbers such that $a/b$ is the probability that $N$ and $n^3-36n$ are relatively prime, find the value of $a+b$.

Prove that for each positive integer $n$ there are at most two pairs $(a, b)$ of positive integers with following two properties: (i) $a^2 + b = n$, (ii) $a+b$ is a power of two, i.e. there is an integer $k \ge 0$ such that $a+b = 2^k$.

A rectangle $ABCD$ has side lengths $AB = m$, $AD = n$, with $m$ and $n$ relatively prime and both odd. It is divided into unit squares and the diagonal AC intersects the sides of the unit squares at the points $A_1 = A, A_2, A_3, \ldots , A_k = C$. Show that \[ A_1A_2 - A_2A_3 + A_3A_4 - \cdots + A_{k-1}A_k = {\sqrt{m^2+n^2}\over mn}. \]

Prove that among any $k + 1$ natural numbers there are two numbers whose difference is divisible by $k$.

Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).

Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is [list=1] [*] 283 [*] 307 [*] 593 [*] 691 [/list]

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

$1019$ stones are placed into two non-empty boxes. Each second Alex chooses a box with an even amount of stones and shifts half of these stones into another box. Prove that for each $k$, $1\le k\le1018$, at some moment there will be a box with exactly $k$ stones. [i](O. Izhboldin)[/i]

Let there be a sequence $a_n$ such that $a_1 = 2,a_2 = 0, a_3 = 1, a_4 = 0$, and for $n \ge 1, a_{n+4}$ is the remainder when $a_n + 2a_{n+1} + 3a_{n+2} + 4a_{n+3}$ is divided by $9$. Prove that there are no positive integer $k$ such that $$a_k = 0, a_{k+1} = 1, a_{k+2} = 0,a_{k+3} = 2.$$

Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ cannot be smaller.

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

There is child camp with some rooms. Call room as $4-$room, if $4$ children live here. Not less then half of all rooms are $4-$rooms , other rooms are $3-$rooms. Not less than $2/3$ girls live in $3-$rooms. Prove that not less than $35\%$ of all children are boys.

Emerald writes $ 2009^2$ integers in a $ 2009\times 2009$ table, one number in each entry of the table. She sums all the numbers in each row and in each column, obtaining $ 4018$ sums. She notices that all sums are distinct. Is it possible that all such sums are perfect squares?

The positive integers $x$ and $y$ are such that $x^{2022}+x+y^2$ is divisible by $xy$. a) Give an example of such integers $x$ and $y$, with $x>y$. b) Prove that $x$ is a perfect square.

[b]p1.[/b] David runs at $3$ times the speed of Alice. If Alice runs $2$ miles in $30$ minutes, determine how many minutes it takes for David to run a mile. [b]p2.[/b] Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$. [b]p3.[/b] Find the $7$-digit palindrome which is divisible by $7$ and whose first three digits are all $2$. [b]p4.[/b] Determine the number of ways to put $5$ indistinguishable balls in $6$ distinguishable boxes. [b]p5.[/b] A certain reduced fraction $\frac{a}{b}$ (with $a,b > 1$) has the property that when $2$ is subtracted from the numerator and added to the denominator, the resulting fraction has $\frac16$ of its original value. Find this fraction. [b]p6.[/b] Find the smallest positive integer $n$ such that $|\tau(n +1)-\tau(n)| = 7$. Here, $\tau(n)$ denotes the number of divisors of $n$. [b]p7.[/b] Let $\vartriangle ABC$ be the triangle such that $AB = 3$, $AC = 6$ and $\angle BAC = 120^o$. Let $D$ be the point on $BC$ such that $AD$ bisect $\angle BAC$. Compute the length of $AD$. [b]p8.[/b] $26$ points are evenly spaced around a circle and are labeled $A$ through $Z$ in alphabetical order. Triangle $\vartriangle LMT$ is drawn. Three more points, each distinct from $L, M$, and $T$ , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap. [b]p9.[/b] Given the three equations $a +b +c = 0$ $a^2 +b^2 +c^2 = 2$ $a^3 +b^3 +c^3 = 19$ find $abc$. [b]p10.[/b] Circle $\omega$ is inscribed in convex quadrilateral $ABCD$ and tangent to $AB$ and $CD$ at $P$ and $Q$, respectively. Given that $AP = 175$, $BP = 147$,$CQ = 75$, and $AB \parallel CD$, find the length of $DQ$. [b]p11. [/b]Let $p$ be a prime and m be a positive integer such that $157p = m^4 +2m^3 +m^2 +3$. Find the ordered pair $(p,m)$. [b]p12.[/b] Find the number of possible functions $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$ that satisfy the following conditions. (1) $f (x) \ne f (y)$ when $x \ne y$ (2) There exists some $x$ such that $f (x)^2 = x^2$ [b]p13.[/b] Let $p$ be a prime number such that there exists positive integer $n$ such that $41pn -42p^2 = n^3$. Find the sum of all possible values of $p$. [b]p14.[/b] An equilateral triangle with side length $ 1$ is rotated $60$ degrees around its center. Compute the area of the region swept out by the interior of the triangle. [b]p15.[/b] Let $\sigma (n)$ denote the number of positive integer divisors of $n$. Find the sum of all $n$ that satisfy the equation $\sigma (n) =\frac{n}{3}$. [b]p16[/b]. Let $C$ be the set of points $\{a,b,c\} \in Z$ for $0 \le a,b,c \le 10$. Alice starts at $(0,0,0)$. Every second she randomly moves to one of the other points in $C$ that is on one of the lines parallel to the $x, y$, and $z$ axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach $(10,10,10)$. [b]p17.[/b] Find the maximum possible value of $$abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3$$ where $a,b,c$ are real such that $a +b +c = 0$. [b]p18.[/b] Circle $\omega$ with radius $6$ is inscribed within quadrilateral $ABCD$. $\omega$ is tangent to $AB$, $BC$, $CD$, and $DA$ at $E, F, G$, and $H$ respectively. If $AE = 3$, $BF = 4$ and $CG = 5$, find the length of $DH$. [b]p19.[/b] Find the maximum integer $p$ less than $1000$ for which there exists a positive integer $q$ such that the cubic equation $$x^3 - px^2 + q x -(p^2 -4q +4) = 0$$ has three roots which are all positive integers. [b]p20.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle ABC = 60^o$,$\angle ACB = 20^o$. Let $P$ be the point such that $CP$ bisects $\angle ACB$ and $\angle PAC = 30^o$. Find $\angle PBC$. PS. You had better use hide for answers.

Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.

Consider the polynomial $P(n) = n^3 -n^2 -5n+ 2$. Determine all integers $n$ for which $P(n)^2$ is a square of a prime. [hide="Remark."]I'm not sure if the statement of this problem is correct, because if $P(n)^2$ be a square of a prime, then $P(n)$ should be that prime, and I don't think the problem means that.[/hide]

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of [i]relatively prime[/i] positive integers: \[f(x+y) = f(x+1) + f(y+1).\]

Find all positive integer solutions $(a, b)$ to the equation $$\frac{1}{a}+\frac{1}{b}+ \frac{n}{lcm(a,b)}=\frac{1}{gcd(a, b)}$$ for (i) $n = 2007$; (ii) $n = 2010$.

Define a [b]ring[/b] in the plane to be the set of points at a distance of at least $r$ and at most $R$ from a specific point $O$, where $r<R$ are positive real numbers. Rings are determined by the three parameters $(O, R, r)$. The area of a ring is labeled $S$. A point in the plane for which both its coordinates are integers is called an integer point. [b]a)[/b] For each positive integer $n$, show that there exists a ring not containing any integer point, for which $S>3n$ and $R<2^{2^n}$. [b]b)[/b] Show that each ring satisfying $100\cdot R<S^2$ contains an integer point.

For a sequence $\{a_i\}_{i\ge1}$ consisting of only positive integers, prove that if for all different positive integers $i$ and $j$, we have $a_i\nmid a_j$, then $$\{p\mid p\text{ is a prime and }p\mid a_i\text{ for some }i\}$$is a infinite set.