Karina, Leticia and Milena paint glass bottles and sell them as decoration. they had $100$ bottles, and they decorated them in such a way that each bottle was painted by a single person. After the finished, they put all the bottles on a table. In an oversight one of them pushed the table, falling and breaking exactly $\frac18$ of the bottles that Karina painted, $\frac13$ of the bottles that Milena, painted and $\frac16$ of the bottles that Leticia painted. In total, $82$ painted bottles remained unbroken. Knowing that the number of broken bottles that Milena had painted is equal to the average of the amounts of broken bottles painted by Karina and Leticia, how many bottles did each of them paint?

Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.

Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.

Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square. Lucian Petrescu

Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.

Define a sequence of integers $a_0=m, a_1=n$ and $a_{k+1}=4a_k-5a_{k-1}$ for all $k \ge 1$. Suppose $p>5$ is a prime with $p \equiv 1 \pmod{4}$. Prove that it is possible to choose $m,n$ such that $p \nmid a_k$ for any $k \ge 0$.

Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.

For a positive integer $n$, the equation $a^2 + b^2 + c^2 + n = abc$ is given in the positive integers. Prove that: 1. There does not exist a solution $(a, b, c)$ for $n = 2017$. 2. For $n = 2016$, $a$ is divisible by $3$ for all solutions $(a, b, c)$. 3. There are infinitely many solutions $(a, b, c)$ for $n = 2016$.

Prove that there are infinitely many different triangles in coordinate plane satisfying: 1) their vertices are lattice points 2) their side lengths are consecutive integers [b]Remark[/b]: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles

A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle.

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

Find all functions $f: \mathbb N \to \mathbb N$ Such that: 1.for all $x,y\in N$:$x+y|f(x)+f(y)$ 2.for all $x\geq 1395$:$x^3\geq 2f(x)$

Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001$.

For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

A natural number $n$ is given. For all integer triplets $(a,b,c)$ such that $0 < |a| , |b|, |c| < 2023$ and satisfying below, show that the product of all possible integer $a$ is a perfect square. (The value of $a$ allows duplication) $$(a+nb)(a-nc) + abc = 0$$

Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

Let $n$ be a positive integer, and let $1=d_1<d_2<\dots < d_k=n$ denote all positive divisors of $n$, If the following conditions are satisfied: $$ 2d_2+d_4+d_5=d_7$$ $$ d_3 d_6 d_7=n$$ $$ (d_6+d_7)^2=n+1$$ find all possible values of $n$.

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

Prove that $1^{2011}+2^{2011}+3^{2011}+...+2012^{2011} $ is divisible by $2025078$.

The natural number $n$ was multiplied by $3$, resulting in the number $999^{1000}$. Find the unity digit of $n$.

Given a positive integer $ n>3$, prove that the least common multiple of the products $ x_1x_2\cdots x_k$ ($ k\geq 1$) whose factors $ x_i$ are positive integers with $ x_1\plus{}x_2\plus{}\cdots\plus{}x_k\le n$, is less than $ n!$.

The sequence $(a_k)$, $k> 0$ is Fibonacci, with $a_0 = a_1 = 1$. Calculate the value of $$\sum_{j = 0}^{\infty} \frac{a_j}{2^j}$$