Find all prime numbers $p$ and $q$ such that $2p^2q + 45pq^2$ is a perfect square.

Call $A \in \mathbb{N}^0$ be $number of year$ if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation). Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.

Solve in prime numbers: $ \{\begin{array}{c}\ \ 2a - b + 7c = 1826 \ 3a + 5b + 7c = 2007 \end{array}$

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

Let $ 1\le a_1<a_2<\cdots<a_m\le N$ be a sequence of integers such that the least common multiple of any two of its elements is not greater than $ N$. Show that $ m\le 2\left[\sqrt{N}\right]$, where $ \left[\sqrt{N}\right]$ denotes the greatest integer $ \le \sqrt{N}$

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine all positive integers whose squares end in $196$.

Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property: for every $x,y\in A$ with $x\neq y$, it holds that \[\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.

Find the greatest positive integer $m$ with the following property: For every permutation $a_1, a_2, \cdots, a_n,\cdots$ of the set of positive integers, there exists positive integers $i_1<i_2<\cdots <i_m$ such that $a_{i_1}, a_{i_2}, \cdots, a_{i_m}$ is an arithmetic progression with an odd common difference.

There are positive integers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ such that $ a+b = c+d = e+f = 101 $. Prove that the number $ \frac{ace}{bdf} $ cannot be written as a fraction $ \frac{m}{n} $ where $ m $, $ n $ are positive integers with a sum less than $ 101 $.

Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not. Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.

Let $P(x)$ be a polynomial with integer coefficients. We denote $\deg(P)$ its degree which is $\geq 1.$ Let $n(P)$ be the number of all the integers $k$ for which we have $(P(k))^{2}=1.$ Prove that $n(P)- \deg(P) \leq 2.$

Determine the values of the positive integer $n$ for which $$A =\sqrt{\frac{9n - 1}{n + 7}}$$ is rational.

Two runners start running along a circular track of unit length from the same starting point and int he same sense, with constant speeds $v_1$ and $v_2$ respectively, where $v_1$ and $v_2$ are two distinct relatively prime natural numbers. They continue running till they simultneously reach the starting point. Prove that (a) at any given time $t$, at least one of the runners is at a distance not more than $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units from the starting point. (b) there is a time $t$ such that both the runners are at least $\frac{[\frac{v_1 + v_2}{2}]}{v_1 + v_2}$ units away from the starting point. (All disstances are measured along the track). $[x]$ is the greatest integer function.

Consider the sequence $x_n>0$ defined with the following recurrence relation: \[x_1 = 0\] and for $n>1$ \[(n+1)^2x_{n+1}^2 + (2^n+4)(n+1)x_{n+1}+ 2^{n+1}+2^{2n-2} = 9n^2x_n^2+36nx_n+32.\] Show that if $n$ is a prime number larger or equal to $5$, then $x_n$ is an integer.

Let $a$, $b$ and $n$ be positive integers, satisfying that $bn$ divides $an-a+1$. Let $\alpha=a/b$. Prove that, when the numbers $\lfloor\alpha\rfloor,\lfloor2\alpha\rfloor,\dots,\lfloor(n-1)\alpha\rfloor$ are divided by $n$, the residues are $1,2,\dots,n-1$, in some order.

Consider the sequence of natural numbers $a_n$ defined as $a_0=4$ and $a_{n+1}=\frac{a_n(a_n-1)}{2}$ for each $n\geq 0$. Define a new sequence $b_n$ as follows: $b_n=0$ if $a_n$ is even, and $b_n=1$ if $a_n$ is odd. Prove that for each natural $m$, the sequence \[b_m, b_{m+1}, b_{m+2},b_{m+3}, \dots\] is not periodic.

Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

Let $ A$ be a subset of the set $ \{1, 2,\ldots,2006\}$, consisting of $ 1004$ elements. Prove that there exist $ 3$ distinct numbers $ a,b,c\in A$ such that $ gcd(a,b)$: a) divides $ c$ b) doesn't divide $ c$

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

How many integers in the set $\{1, 2 ,..., 2010\}$ divide $5^{2010!}- 3^{2010!}$?

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

Find all triplets (a, b, c) of positive integers so that $a^2b$, $b^2c$ and $c^2a$ devide $a^3+b^3+c^3$

Find the greatest real number $ \alpha$ for which there exists a sequence of infinitive integers $ (a_n)$, ($ n \equal{} 1, 2, 3, \ldots$) satisfying the following conditions: 1) $ a_n > 1997n$ for every $ n \in\mathbb{N}^{*}$; 2) For every $ n\ge 2$, $ U_n\ge a^{\alpha}_n$, where $ U_n \equal{} \gcd\{a_i \plus{} a_k | i \plus{} k \equal{} n\}$.

Let $f_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}$. Find $\lim \limits_{n \to \infty}(f_{n+1} - f_n)$ where $F_n$ denotes the $n$th Fibonacci number (given by $F_0 = 0$, $F_1 = 1$, and by $F_{n+1} = F_n + F_{n-1}$ for all $n \geq 1$