Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.

$a_1,…,a_n$ are real numbers such that $a_1+…+a_n=0$ and $|a_1|+…+|a_n|=1$. Prove that : $$|a_1+2a_2+…+na_n| \leq \frac{n-1}{2}$$

$ABC$ is a non-isosceles triangle. $O, I, H$ are respectively the center of its circumscribed circle, the inscribed circle and its orthocenter. prove that $\widehat{OIH}$ is obtuse.

Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.

Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square

Find the maximal value of the following expression, if $a,b,c$ are nonnegative and $a+b+c=1$. \[ \frac{1}{a^2 -4a+9} + \frac {1}{b^2 -4b+9} + \frac{1}{c^2 -4c+9} \]

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. Proposed by Ismail Isaev and Mikhail Isaev, Russia