Olympiad problems

1996 Tournament Of Towns, (504) 1

Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers? (Inspired by a diagram in an old text book)

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 $3$.

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)

2018 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: sum of cubes , perfect square , perfect cube , sum of squares , number theory

Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.

2001 Romania Team Selection Test, 4

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

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

1995 Denmark MO - Mohr Contest, 2

Tags: algebra , sum of squares

Find all sets of five consecutive integers with that property that the sum of the squares of the first three numbers is equal to the sum of the squares on the last two.

1989 Austrian-Polish Competition, 9

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

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

2001 Singapore MO Open, 4

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

A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer. (As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).

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 $13^k$.

2017 Singapore MO Open, 3

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

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

2014 IFYM, Sozopol, 3

Tags: polynomial , algebra , sum of squares , number theory

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]

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 \[x^2+y^2=3z^2.\]

2021 Poland - Second Round, 3

Tags: sum of squares , number theory

Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

2006 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , sum of squares , perfect square

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

2010 IMO Shortlist, 3

Tags: algebra , number theory , polynomial , 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]

2010 Saudi Arabia Pre-TST, 2.2

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

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

2010 Greece JBMO TST, 1

Tags: algebra , number theory , sum of squares , digit

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

1976 Dutch Mathematical Olympiad, 1

Tags: number theory , sum of squares

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

2010 Contests, 1

Tags: algebra , number theory , sum of squares , digit

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

2018 Romania Team Selection Tests, 2

Tags: number theory , quadratic reciprocity , sum of squares , composite number

Show that a number $n(n+1)$ where $n$ is positive integer is the sum of 2 numbers $k(k+1)$ and $m(m+1)$ where $m$ and $k$ are positive integers if and only if the number $2n^2+2n+1$ is composite.

2013 Bosnia and Herzegovina Junior BMO TST, 1

Tags: divisible , number theory , sum of squares

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

1978 IMO Shortlist, 17

Tags: number theory , quadratic , 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.

1978 IMO Longlists, 17

Tags: number theory , quadratic , 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.

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?

2015 Thailand TSTST, 1

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

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.