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

1981 Spain Mathematical Olympiad, 8
Tags: sum of squares , divisible , number theory
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 $4$.

1985 All Soviet Union Mathematical Olympiad, 396
Tags: sum of digits , sum of squares , sum , decimal representation , digit , number theory
Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

2022 Durer Math Competition Finals, 8
Tags: algebra , number theory , sum of squares
The product of Albrecht’s three favorite numbers is $2022$, and if we add one to each number, their product will be $1514$. What is the sum of their squares, if we know their sum is $0$?

1991 ITAMO, 3
Tags: number theory , sum , sum of squares
We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)

1966 Swedish Mathematical Competition, 3
Tags: number theory , sum of squares , perfect square
Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

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?

2021 South Africa National Olympiad, 3
Tags: number theory , prime , sum of squares , power of 2 , perfect square
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2013 Saudi Arabia BMO TST, 5
Tags: sum of squares , digit , perfect square , number theory
We call a positive integer [i]good[/i ] if it doesn’t have a zero digit and the sum of the squares of its digits is a perfect square. For example, $122$ and $34$ are good and $304$ and $12$ are not not good. Prove that there exists a $n$-digit good number for every positive integer $n$.

1987 All Soviet Union Mathematical Olympiad, 441
Tags: algebra , combinatorics , sum of squares , tournament
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 $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

2000 IMO Shortlist, 6
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.