Olympiad problems

1995 Denmark MO - Mohr Contest, 2

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.

2001 Singapore MO Open, 4

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

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 $5^2 = 3^2 + 4^2$).

2018 Rioplatense Mathematical Olympiad, Level 3, 1

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

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: 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.

2021 South Africa National Olympiad, 3

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

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

2018 Romania National Olympiad, 1

Tags: number theory , sum of squares

Find the distinct positive integers $a, b, c,d$, such that the following conditions hold: (1) exactly three of the four numbers are prime numbers; (2) $a^2 + b^2 + c^2 + d^2 = 2018.$

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

2014 IFYM, Sozopol, 3

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

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]

2022 Durer Math Competition Finals, 8

Tags: number theory , sum of squares , algebra

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 $0$?

1949-56 Chisinau City MO, 8

Tags: number theory , sum of squares , remainder

Prove that the remainder of dividing the sum of two squares of integers by $4$ is different from $3$.

1989 Austrian-Polish Competition, 9

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

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.

2017 Thailand Mathematical Olympiad, 5

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

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.

2000 ITAMO, 1

Tags: consecutive , sum of squares , number theory

A possitive integer is called [i]special[/i] if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$-digit [i]special[/i] numbers (b) Are there $2000$-digit [i]special[/i] numbers?

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 $n$.

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.

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.

2022 JBMO Shortlist, N6

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

Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.

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.

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)

1966 Swedish Mathematical Competition, 3

Tags: sum of squares , number theory , perfect square

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

1987 All Soviet Union Mathematical Olympiad, 441

Tags: sum of squares , algebra , combinatorics , 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 $$x_1^2 + ... + x_{10}^2 = y_1^2 + ... + y_{10}^2$$

2010 IMO Shortlist, 3

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