Let $0<x<y<z<p$ be integers where $p$ is a prime. Prove that the following statements are equivalent: $(a) x^3\equiv y^3\pmod p\text{ and }x^3\equiv z^3\pmod p$ $(b) y^2\equiv zx\pmod p\text{ and }z^2\equiv xy\pmod p$

Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio $3: 2: 1$, what is the least possible total for the number of bananas?

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

In a football season, even number $n$ of teams plays a simple series, i.e. each team plays once against each other team. Show that ona can group the series into $n-1$ rounds such that in every round every team plays exactly one match.

Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions (a) $0\leq A<B<C\leq99$, (b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\leq b < a < c < p$, (c) $p$ divides $A-a$, $B-b$, and $C-c$, and (d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences. Find $N$.

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

Let $a,b$ be two integers such that their gcd has at least two prime factors. Let $S = \{ x \mid x \in \mathbb{N}, x \equiv a \pmod b \} $ and call $ y \in S$ irreducible if it cannot be expressed as product of two or more elements of $S$ (not necessarily distinct). Show there exists $t$ such that any element of $S$ can be expressed as product of at most $t$ irreducible elements.

Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there a) exist b) exist infinitely many $x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$. (proposed by B. Bayarjargal)

Let $ a>1$ be a positive integer. Prove that every non-zero positive integer $ N$ has a multiple in the sequence $ (a_n)_{n\ge1}$, $ a_n\equal{}\left\lfloor\frac{a^n}n\right\rfloor$.

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k,\thinspace 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: $\bullet$ Any cube may be the bottom cube in the tower. $\bullet$ The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

What is the smallest prime number dividing the sum $ 3^{11} \plus{} 5^{13}$? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 3^{11} \plus{} 5^{13}\qquad\textbf{(E)}\ \text{none of these}$

How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$?

For any nonnegative integer $ n$, let $ f(n)$ be the greatest integer such that $ 2^{f(n)} | n \plus{} 1$. A pair $ (n, p)$ of nonnegative integers is called nice if $ 2^{f(n)} > p$. Find all triples $ (n, p, q)$ of nonnegative integers such that the pairs $ (n, p)$, $ (p, q)$ and $ (n \plus{} p \plus{} q, n)$ are all nice.

Prove that there are no positive integers $a,b$ such that $(15a +b)(a +15b)$ is a power of $3.$

Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]

In year $ N$, the $ 300^\text{th}$ day of the year is a Tuesday. In year $ N \plus{} 1$, the $ 200^{\text{th}}$ day of the year is also a Tuesday. On what day of the week did the $ 100^\text{th}$ day of year $ N \minus{} 1$ occur? $ \textbf{(A)}\ \text{Thursday} \qquad \textbf{(B)}\ \text{Friday} \qquad \textbf{(C)}\ \text{Saturday} \qquad \textbf{(D)}\ \text{Sunday} \qquad \textbf{(E)}\ \text{Monday}$

Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and \[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]

Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

[b]a)[/b] Find all prime numbers $p, q, r$ satisfying $3 \nmid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares. [b]b)[/b] Do there exist prime numbers $p, q, r$ such that $3 \mid p+q+r$ and $p+q+r$ and $pq+qr+rp+3$ are both perfect squares?

Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$

For a positive integer, we define it's [i]set of exponents[/i] the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$). [i]Proposed by A. Golovanov[/i]

For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$