For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system. Show that there is an integer $k$, with no $9$ in it's decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. [i]Proposed by Yinghua Ai[/i]

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

Determine all pairs $(m, n)$ of positive integers for which $(m + n)^3 / 2n (3m^2 + n^2) + 8$

Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

Let $\lfloor z \rfloor$ denote the greatest integer less than or equal to $z.$ Compute $$\sum_{j=-1000}^{1000} \left\lfloor \frac{2025}{j+0.5}\right\rfloor.$$

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

Assume that the set of all positive integers is decomposed into $ r$ (disjoint) subsets $ A_1 \cup A_2 \cup \ldots \cup A_r \equal{} \mathbb{N}.$ Prove that one of them, say $ A_i,$ has the following property: There exists a positive $ m$ such that for any $ k$ one can find numbers $ a_1, a_2, \ldots, a_k$ in $ A_i$ with $ 0 < a_{j \plus{} 1} \minus{} a_j \leq m,$ $ (1 \leq j \leq k \minus{} 1)$.

Given is the equation $(m, n) +[m, n] =m+n$ where $m, n$ are positive integers and m>n. a) Prove that n divides m. b) If $m-n=10$, solve the equation.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and \[x^3(y^3+z^3)=2012(xyz+2).\]

Let $a_n$ be a sequence of natural numbers defined by $a_1 = m$ and for $n > 1$. We call apair$ (a_k, a_{\ell })$ [i]interesting [/i] if (i) $0 < \ell - k < 2016$, (ii) $a_k$ divides $a_{\ell }$. Show that there exists a $m$ such that the sequence $a_n$ contains no interesting pair.

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

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.

[b]7.1 . [/b] The area of the quadrilateral is $3$ cm$^2$ , and the lengths of its diagonals are $6$ cm and $2$ cm. Find the angle between the diagonals. [b]7.2[/b] Prove that the number $1 + 2^{3456789}$ is composite. [b]7.3[/b] $20$ people took part in the chess tournament. The participant who took clear (undivided) $19$th place scored $9.5$ points. How could they distribute points among other participants? [b]7.4[/b] The sum of the distances between the midpoints of opposite sides of a quadrilateral is equal to its semi-perimeter. Prove that this quadrilateral is a parallelogram. [b]7.5[/b] $40$ people travel on a bus without a conductor passengers carrying only coins in denominations of $10$, $15$ and $20$ kopecks. Total passengers have $ 49$ coins. Prove that passengers will not be able to pay the required amount of money to the ticket office and pay each other correctly. (Cost of a bus ticket in 1963 was 5 kopecks.) [b]7.6[/b] Some natural number $a$ is divided with a remainder by all natural numbers less than $a$. The sum of all the different (!) remainders turned out to be equal to $a$. Find $a$. [b]7.7[/b] Two squares were cut out of a chessboard. In what case is it possible and in what case not to cover the remaining squares of the board with dominoes (i.e., figures of the form $2\times 1$) without overlapping? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$. (a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced. (b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$. [i]Proposed by Jorge Tipe, Peru[/i]

An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.

Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

Given $8$ pairwise distinct positive integers $a_1,a_2,…,a_8$ such that the greatest common divisor of any three of them is equal to $1$. Show that there exists positive integer $n\geq 8$ and $n$ pairwise distinct positive integers $m_1,m_2,…,m_n$ with the greatest common divisor of all $n$ numbers equal to $1$ such that for any positive integers $1\leq p<q<r\leq n$, there exists positive integers $1\leq i<j\leq 8$ that $a_ia_j\mid m_p+m_q+m_r$.