Richard and Kaarel are taking turns to choose numbers from the set $\{1,\dots,p-1\}$ where $p > 3$ is a prime. Richard is the first one to choose. A number which has been chosen by one of the players cannot be chosen again by either of the players. Every number chosen by Richard is multiplied with the next number chosen by Kaarel. Kaarel wins the game if at any moment after his turn the sum of all of the products calculated so far is divisible by $p$. Richard wins if this does not happen, i.e. the players run out of numbers before any of the sums is divisible by $p$. Can either of the players guarantee their victory regardless of their opponent's moves and if so, which one?

Given is a positive integer $n$ divisible by $3$ and such that $2n-1$ is a prime. Does there exist a positive integer $x>n$ such that $$nx^{n+1}+(2n+1)x^n-3(n-1)x^{n-1}-x-3$$ is a product of the first few odd primes?

Exists a positive integer $n$ such that the number $\underbrace{1...1}_{n \,ones} 2 \underbrace{1...1}_{n \, ones}$ is a prime number?

a) Prove that if $n$ is an odd perfect number then $n$ has the following form \[ n=p^sm^2 \] where $p$ is prime has form $4k+1$, $s$ is positive integers has form $4h+1$, and $m\in\mathbb{Z}^+$, $m$ is not divisible by $p$. b) Find all $n\in\mathbb{Z}^+$, $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number

Two positive integers $m$ and $n$ satisfy $$max \,(m, n) = (m - n)^2$$ $$gcd \,(m, n) = \frac{min \,(m, n)}{6}$$ Find $lcm\,(m, n)$

Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \][i]Proposed by Aaron Lin[/i]

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

The world-renowned Marxist theorist [i]Joric[/i] is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer $n$, he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the [i]defect[/i] of the number $n$. Determine the average value of the defect (over all positive integers), that is, if we denote by $\delta(n)$ the defect of $n$, compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\] [i]Iurie Boreico[/i]

Is it possible to complete the following square knowning that each row and column make an aritmetic progression?

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

Let $(F_n)_{n\in\mathbb{N}}$ be the Fibonacci sequence difined as $F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}$. Show that for every nonnegative integer $r$ there is a term in the Fibonacci sequence that is divided by $r$.

2. Prove that for any $2\le n \in \mathbb{N}$ there exists positive integers $a_1,a_2,\cdots,a_n$ such that $\forall i\neq j: \text{gcd}(a_i,a_j) = 1$ and $\forall i: a_i \ge 1402$ and the given relation holds. $$[\frac{a_1}{a_2}]+[\frac{a_2}{a_3}]+\cdots+[\frac{a_n}{a_1}] = [\frac{a_2}{a_1}]+[\frac{a_3}{a_2}]+\cdots+[\frac{a_1}{a_n}]$$

Determine all ten-digit numbers whose decimal $\overline{a_0a_1a_2a_3a_4a_5a_6a_7a_8a_9}$ is given by such that for each integer $j$ with $0\le j \le 9, a_j$ is equal to the number of digits equal to $j$ in this representation. That is: the first digit is equal to the amount of "$0$" in the writing of that number, the second digit is equal to the amount of "$1$" in the writing of that number, the third digit is equal to the amount of "$2$" in the writing of that number, ... , the tenth digit is equal to the number of "$9$" in the writing of that number.

If $ \frac{a }{b}+ \frac{b}{c}+ \frac{c}{a}$ is integer. show that $ abc$ is perfect cube.

For what values of $k\ge2$ can the set of natural numbers be colored in $k$ colors in such a way that it contains no single - color infinite arithmetic progression, but for any two colors there is a progression whose members are each colored in one of these two colors?

Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions: $i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$ $ii) f$ takes all integer values

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

Prove that among any $12$ consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers other than $1$ and $n$ which divide $n$. For example, the proper divisors of $14$ are $2$ and $7$.)

Let $S(x)$ denote the sum of the decimal digits of $x$. Do there exist natural numbers $a,b,c$ such that \[ S(a+b)<5, \quad S(b+c)<5, \quad S(c+a)<5, \quad S(a+b+c)> 50? \]

Let $k>1$ and $n$ be natural numbers and $p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$. Prove that, if $p$ is prime, then $n|k!$.

Triangle $A_1B_1C_1$ is an equilateral triangle with sidelength $1$. For each $n>1$, we construct triangle $A_nB_nC_n$ from $A_{n-1}B_{n-1}C_{n-1}$ according to the following rule: $A_n,B_n,C_n$ are points on segments $A_{n-1}B_{n-1},B_{n-1}C_{n-1},C_{n-1}A_{n-1}$ respectively, and satisfy the following: \[\dfrac{A_{n-1}A_n}{A_nB_{n-1}}=\dfrac{B_{n-1}B_n}{B_nC_{n-1}}=\dfrac{C_{n-1}C_n}{C_nA_{n-1}}=\dfrac1{n-1}\] So for example, $A_2B_2C_2$ is formed by taking the midpoints of the sides of $A_1B_1C_1$. Now, we can write $\tfrac{|A_5B_5C_5|}{|A_1B_1C_1|}=\tfrac mn$ where $m$ and $n$ are relatively prime integers. Find $m+n$. (For a triangle $\triangle ABC$, $|ABC|$ denotes its area.)

Let $x$ satisfy $(6x + 7) + (8x + 9) = (10 + 11x) + (12 + 13x).$ There are relatively prime positive integers so that $x = -\tfrac{m}{n}$. Find $m + n.$

[i](6 pts)[/i] Find all positive integers $n \ge 3$ such that there exists a permutation $a_{1}, a_{2}, \dots, a_{n}$ of $1, 2, \dots, n$ such that $a_{1}, 2a_{2}, \dots, na_{n}$ can be rearranged into an arithmetic progression.

For each positive integer $n$ let the set $A_n$ consist of all numbers $\pm 1 \pm 2 \pm ...\pm n$. For example, $$A_1 = \{-1,1\}, A_2 = \{ -3 ,-1 ,1 ,3 \} , A_3 = \{ -6 ,-4 ,-2 ,0 ,2 ,4 ,6 \}.$$ Find the number of elements in $A_n$ .

Professor Conway collects a total of $58$ midterms from the two sections of his introductory linear algebra course. He notices that the number of midterms from the smaller section is equal to the product of the digits of the number of midterms from his larger section. Assuming that every student handed in a midterm, how many students are there in the smaller section?