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: 60

2014 Federal Competition For Advanced Students, P2, 4
Tags: number theory , sum of squares , product , divisor , consecutive
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?

1974 Swedish Mathematical Competition, 6
Tags: number theory , perfect square , sum of squares
For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that \[ a_1^2+a_2^2+\cdots+a_n^2 \] is a square?

2013 China Northern MO, 4
Tags: number theory , sum of squares
For positive integers $n,a,b$, if $n=a^2 +b^2$, and $a$ and $b$ are coprime, then the number pair $(a,b)$ is called a [i]square split[/i] of $n$ (the order of $a, b$ does not count). Prove that for any positive $k$, there are only two square splits of the integer $13^k$.

2025 Macedonian Balkan MO TST, 4
Tags: number theory , sum of squares , prime
Let $n$ be a positive integer. Prove that for every odd prime $p$ dividing $n^2 + n + 2$, there exist integers $a, b$ such that $p = a^2 + 7b^2$.

2018 Romania National Olympiad, 1
Tags: perfect square , sum of squares , number theory
Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.

2015 Bosnia Herzegovina Team Selection Test, 4
Tags: sum of squares , set , combinatorics
Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.

2010 Saudi Arabia Pre-TST, 2.2
Tags: prime , consecutive , number theory , sum of squares
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

1999 Tournament Of Towns, 3
Tags: number theory , divide , divisible , sum of squares
Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

1981 Spain Mathematical Olympiad, 8
Tags: number theory , sum of squares , divisible
If $a$ is an odd number, show that $$a^4 + 4a^3 + 11a^2 + 6a+ 2$$ is a sum of three squares and is divisible by $4$.

2019 Brazil Team Selection Test, 4
Tags: number theory , prime number , sum of squares , set
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.