Let $x,y$ be positive integers such that $3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

Evaluate the sum of all positive irreducible fractions less than $1$ and having the denominator $1991$.

An infinite checkerboard is divided by a horizontal line into upper and lower halves as shown on the right. A number of checkers are to be placed on the board below the line (within the squares). A "move" consists of one checker jumping horizontally or vertically over a second checker, and removing the second checker. What is the minimum value of $n$ which will allow the placement of the last checker in row 4 above the dividing horizontal line after $n-1$ moves? Describe the initial position of the checkers as well as each of the moves. Picture: http://www.cms.math.ca/Competitions/IMTS/imts6.gif

Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\] true?

Let $b\ge 2$ be an integer. Call a positive integer $n$ $b$-[i]eautiful[/i] if it has exactly two digits when expressed in base $b$ and these two digits sum to $\sqrt{n}$. For example, $81$ is $13$-[i]eautiful[/i] because $81 = \underline{6} \ \underline{3}_{13} $ and $6 + 3 = \sqrt{81}$. Find the least integer $b\ge 2$ for which there are more than ten $b$-[i]eautiful[/i] integers.

Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$.

Prove that one can arrange all positive divisors of any given positive integer around a circle so that for any two neighboring numbers one is divisible by another.

Let $a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3,\ldots$ be positive integers such that $a_{n+2} = a_n + a_{n+1}$ and $b_{n+2} = b_n + b_{n+1}$ for all $n \ge 1$. Assume that $a_n$ divides $b_n$ for infinitely many values of $n$. Prove that there exists an integer $c$ such that $b_n = c a_n$ for all $n \ge 1$.

Describe all the ways to color each natural number as one of three colors so that the following condition is satisfied: if the numbers $a$, $b$ and $c$ (not necessarily different) satisfy the condition $2000(a + b) = c$, then they either all the same color or three different colors

Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

Let $a$ and $n$ be positive integers and $s = a + a^2 + \cdots + a^n$. Prove that the last digit of $s$ is $1$ if and only if the last digits of $a$ and $n$ are both equal to $1$.

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\dots 111$. Compute the sum of the digits of the number $9L^2+2L$.

A prime is called [i]perfect[/i] if there is a permutation $a_1, a_2, \cdots, a_{\frac{p-1}{2}}, b_1, b_2, \cdots, b_{\frac{p-1}{2}}$ of $1, 2, \cdots, p-1$ satisfies $$b_i \equiv a_i + \frac{1}{a_i} \pmod p$$ for all $1 \le i \le \frac{p-1}{2}$. Show that there are infinitely many primes that are not perfect. [i]Proposed By - CSJL[/i]

Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Determine all integers $n>1$ that satisfy the following condition: for any positive integer $x$, if gcd$(x,n)=1$, then gcd$(x+101,n)=1$.

Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" . (1) Is $(s,t)$ a good pair ? (2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".

Find all triplets of positive integers $(a,b,c)$ for which the number $3^a+3^b+3^c$ is a perfect square.

Let s be a positive real. Consider a two-dimensional Cartesian coordinate system. A [i]lattice point[/i] is defined as a point whose coordinates in this system are both integers. At each lattice point of our coordinate system, there is a lamp. Initially, only the lamp in the origin of the Cartesian coordinate system is turned on; all other lamps are turned off. Each minute, we additionally turn on every lamp L for which there exists another lamp M such that - the lamp M is already turned on, and - the distance between the lamps L and M equals s. Prove that each lamp will be turned on after some time ... [b](a)[/b] ... if s = 13. [This was the problem for class 11.] [b](b)[/b] ... if s = 2005. [This was the problem for classes 12/13.] [b](c)[/b] ... if s is an integer of the form $s=p_1p_2...p_k$ if $p_1$, $p_2$, ..., $p_k$ are different primes which are all $\equiv 1\mod 4$. [This is my extension of the problem, generalizing both parts [b](a)[/b] and [b](b)[/b].] [b](d)[/b] ... if s is an integer whose prime factors are all $\equiv 1\mod 4$. [This is ZetaX's extension of the problem, and it is stronger than [b](c)[/b].] Darij

The positive integers $x_1, \cdots , x_n$, $n \geq 3$, satisfy $x_1 < x_2 <\cdots< x_n < 2x_1$. Set $P = x_1x_2 \cdots x_n.$ Prove that if $p$ is a prime number, $k$ a positive integer, and $P$ is divisible by $pk$, then $\frac{P}{p^k} \geq n!.$