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

1987 All Soviet Union Mathematical Olympiad, 441
Tags: sum of squares , combinatorics , tournament , algebra
Ten sportsmen have taken part in a table-tennis tournament (each pair has met once only, no draws). Let $xi$ be the number of $i$-th player victories, $yi$ -- losses. Prove that $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

2016 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: game strategy , combinatorics , sum of squares , game
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

1976 Czech and Slovak Olympiad III A, 1
Tags: diophantine equation , perfect square , sum of squares , number theory
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

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 $13^k$.

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 $4$.

2014 IFYM, Sozopol, 3
Tags: algebra , polynomial , number theory , sum of squares
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\] [i]Proposed by Mariusz Skałba, Poland[/i]

1989 All Soviet Union Mathematical Olympiad, 494
Tags: sum of squares , number theory , 5-digit , algebra , subset
Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

2017 Regional Competition For Advanced Students, 4
Tags: number theory , sum of squares , divisibility , divisor
Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

2015 Thailand TSTST, 1
Tags: number theory , square , sum of squares , perfect square
Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

2001 Romania Team Selection Test, 4
Tags: number theory , additive number theory , sum of squares , perfect square
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.