This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 2

2022 Kazakhstan National Olympiad, 2
Tags: Kazakhstan , Kazakhstan 2022 , number theory proposed , number theory , prime numbers
Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

2022 Kazakhstan National Olympiad, 5
Tags: inequalities , Kazakhstan 2022 , inequalities proposed , algebra
For positive reals $a,b,c$ with $\sqrt{a}+\sqrt{b}+\sqrt{c}\ge 3$ prove that $$\frac{a^3}{a^2+b}+\frac{b^3}{b^2+c}+\frac{c^3}{c^2+a}\ge \frac{3}{2}$$