Prove that there exist infinitely many positive integers $n$ such that $3^n+2$ and $5^n+2$ are all composite numbers.

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

Let $a_0, a_1, \ldots, a_9$ and $b_1 , b_2, \ldots,b_9$ be positive integers such that $a_9<b_9$ and $a_k \neq b_k, 1 \leq k \leq 8.$ In a cash dispenser/automated teller machine/ATM there are $n\geq a_9$ levs (Bulgarian national currency) and for each $1 \leq i \leq 9$ we can take $a_i$ levs from the ATM (if in the bank there are at least $a_i$ levs). Immediately after that action the bank puts $b_i$ levs in the ATM or we take $a_0$ levs. If we take $a_0$ levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of $n$ such that after finite number of takings money from the ATM there will be no money in it.

Let $ n$ be a natural number, such that $ (n,2(2^{1386}\minus{}1))\equal{}1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:\[ n|a_{1}^{1386}\plus{}a_{2}^{1386}\plus{}\dots\plus{}a_{\varphi(n)}^{1386}\]

Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.

Determine all integral solutions of \[ a^2\plus{}b^2\plus{}c^2\equal{}a^2b^2.\]

Find all positive integers $n$ for which there exist two distinct numbers of $n$ digits, $\overline{a_1a_2\ldots a_n}$ and $\overline{b_1b_2\ldots b_n}$, such that the number of $2n$ digits $\overline{a_1a_2\ldots a_nb_1b_2\ldots b_n}$ is divisible by $\overline{b_1b_2\ldots b_na_1a_2\ldots a_n}$.

Find all pairs $\left(m,n\right)$ of positive integers, with $m,n\geq2$, such that $a^n-1$ is divisible by $m$ for each $a\in \left\{1,2,3,\ldots,n\right\}$.

Find all positive integers $m,n$ and prime numbers $p$ for which $\frac{5^m+2^np}{5^m-2^np}$ is a perfect square.

Let $p>3$ is a prime number and $k=\lfloor\frac{2p}{3}\rfloor$. Prove that \[{p \choose 1}+{p \choose 2}+\cdots+{p \choose k}\] is divisible by $p^{2}$.

What is the last two digits of the decimal representation of $9^{8^{7^{\cdot^{\cdot^{\cdot^{2}}}}}}$? $ \textbf{(A)}\ 81 \qquad\textbf{(B)}\ 61 \qquad\textbf{(C)}\ 41 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 01 $

A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ [i]Dan Schwarz, Romania[/i]

Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.

A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors). Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.) [hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]

Find all triples $(p,q,n)$ that satisfy \[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\] where $p,q$ are odd primes and $n$ is an positive integer.

This is an ARML Super Relay! I'm sure you know how this works! You start from #1 and #15 and meet in the middle. We are going to require you to solve all $15$ problems, though -- so for the entire task, submit the sum of all the answers, rather than just the answer to #8. Also, uhh, we can't actually find the slip for #1. Sorry about that. Have fun anyways! Problem 2. Let $T = TNYWR$. Find the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Problem 3. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 4. Let $T = TNYWR$ and flip $4$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 5. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 6. Let $T = TNYWR$ and flip $6$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 7. Let $T = TNYWR$. Compute the smallest prime $p$ for which $n^T \not\equiv n \pmod{p}$ for some integer $n$. Problem 8. Let $M$ and $N$ be the two answers received, with $M \le N$. Compute the number of integer quadruples $(w,x,y,z)$ with $w+x+y+z = M \sqrt{wxyz}$ and $1 \le w,x,y,z \le N$. Problem 9. Let $T = TNYWR$. Compute the smallest integer $n$ with $n \ge 2$ such that $n$ is coprime to $T+1$, and there exists positive integers $a$, $b$, $c$ with $a^2+b^2+c^2 = n(ab+bc+ca)$. Problem 10. Let $T = TNYWR$ and flip $10$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 11. Let $T = TNYWR$. Compute the last digit of $T^T$ in base $10$. Problem 12. Let $T = TNYWR$ and flip $12$ fair coins. Suppose the probability that at most $T$ heads appear is $\frac mn$, where $m$ and $n$ are coprime positive integers. Compute $m+n$. Problem 13. Let $T = TNYWR$. If $d$ is the largest proper divisor of $T$, compute $\frac12 d$. Problem 14. Let $T = TNYWR$. Compute the number of way to distribute $6$ indistinguishable pieces of candy to $T$ hungry (and distinguishable) schoolchildren, such that each child gets at most one piece of candy. Also, we can't find the slip for #15, either. We think the SFBA coaches stole it to prevent us from winning the Super Relay, but that's not going to stop us, is it? We have another #15 slip that produces an equivalent answer. Here you go! Problem 15. Let $A$, $B$, $C$ be the answers to #8, #9, #10. Compute $\gcd(A,C) \cdot B$.

What is the sum of distinct remainders when $(2n-1)^{502}+(2n+1)^{502}+(2n+3)^{502}$ is divided by $2012$ where $n$ is positive integer? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 1510 \qquad \textbf{(C)}\ 1511 \qquad \textbf{(D)}\ 1514 \qquad \textbf{(E)}\ \text{None}$

Let $ p>2 $ be a prime number. For any permutation $ \pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) ) $ of the set $ S = \{ 1, 2, \cdots , p \} $, let $ f( \pi ) $ denote the number of multiples of $ p $ among the following $ p $ numbers: \[ \pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p) \] Determine the average value of $ f( \pi) $ taken over all permutations $ \pi $ of $ S $.

Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

Let $n$ be an odd integer greater than 1 and let $c_1, c_2, \ldots, c_n$ be integers. For each permutation $a = (a_1, a_2, \ldots, a_n)$ of $\{1,2,\ldots,n\}$, define $S(a) = \sum_{i=1}^n c_i a_i$. Prove that there exist permutations $a \neq b$ of $\{1,2,\ldots,n\}$ such that $n!$ is a divisor of $S(a)-S(b)$.

For any positive integer $n$ , Prove that there is not exist three odd integer $x,y,z$ satisfing the equation $(x+y)^n+(y+z)^n=(x+z)^n$.

Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and \[1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 \]

Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$

Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]