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

Tags were heavily modified to better represent problems.

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Found problems: 80

Kvant 2021, M2657
Tags: number theory , Russia , All Russian Olympiad
Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.

2021 All-Russian Olympiad, 7
Tags: number theory , Russia , All Russian Olympiad
Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.

Kvant 2023, M2740
Tags: number theory , Russia
Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

1995 All-Russian Olympiad, 6
Tags: geometry , circumcircle , Russia
Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right. [i]L. Kuptsov[/i]

1962 All-Soviet Union Olympiad, 14
Tags: combinatorics , Russia
Given are two sets of positive numbers with the same sum. The first set has $m$ numbers and the second $n$. Prove that you can find a set of less than $m+n$ positive numbers which can be arranged to part fill an $m \times n$ array, so that the row and column sums are the two given sets.