Olympiad problems

1974 Swedish Mathematical Competition, 6

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?

2006 All-Russian Olympiad Regional Round, 11.7

Tags: perfect square , sum of squares , number theory

Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.

1978 IMO Shortlist, 17

Tags: quadratic , number theory , prime , sum of squares

Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$. Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.

2015 Bosnia Herzegovina Team Selection Test, 4

Tags: combinatorics , sum of squares , set

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 $B$.

1987 Tournament Of Towns, (141) 1

Tags: number theory , sum of squares , perfect square , odd

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2016 India Regional Mathematical Olympiad, 5

Tags: number theory , sum of squares

a.) A 7-tuple $(a_1,a_2,a_3,a_4,b_1,b_2,b_3)$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \le i<j \le 4$ and $1 \le k \le 3$, $a_i^2+a_j^2 \not= b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.

2011 QEDMO 8th, 2

Tags: number theory , sum of squares

Let $n$ be an integer. Show that a natural number $k$ can be found for which, the following applies with a suitable choice of signs: $$n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2$$

2021 South Africa National Olympiad, 3

Tags: number theory , sum of squares , power of 2 , prime , perfect square

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

1976 Dutch Mathematical Olympiad, 1

Tags: sum of squares , number theory

Prove that there is no natural $n$ such that $8n + 7$ is the sum of three squares.

2013 Saudi Arabia BMO TST, 5

Tags: number theory , sum of squares , digit , perfect square

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 $n$.

1966 Swedish Mathematical Competition, 3

Tags: sum of squares , perfect square , number theory

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

1952 Moscow Mathematical Olympiad, 216

Tags: integer sequence , sequence , sum of squares , digit , number theory

A sequence of integers is constructed as follows: $a_1$ is an arbitrary three-digit number, $a_2$ is the sum of squares of the digits of $a_1, a_3$ is the sum of squares of the digits of $a_2$, etc. Prove that either $1$ or $4$ must occur in the sequence $a_1, a_2, a_3, ....$

1991 ITAMO, 3

Tags: sum , sum of squares , number theory

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 $8 = 1-4-9+16+25-36-49+64$.)

2017 Singapore MO Open, 3

Tags: integer , number theory , sum , sum of squares

Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.

2013 Bosnia and Herzegovina Junior BMO TST, 1

Tags: sum of squares , divisible , number theory

It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers

1991 Mexico National Olympiad, 5

Tags: consecutive , sum of squares , number theory

The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$. Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.

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 $p = a^2 + 7b^2$.

2006 Kazakhstan National Olympiad, 7

Tags: sum of squares , number theory

Prove that if a natural number $ N $ can be represented in the form the sum of three squares of integers divisible by $3$, then it is also is represented as the sum of three squares of integers that are not divisible by $3$.

2019 Tournament Of Towns, 4

Tags: number theory , consecutive , multiple , sum of squares , circles

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)