Olympiad problems

2000 IMO Shortlist, 6

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

1985 All Soviet Union Mathematical Olympiad, 396

Tags: sum of digits , sum of squares , sum , number theory , decimal representation , digit

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 $1000000$?

2019 Brazil Team Selection Test, 4

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

Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.

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.

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) $a^2 + b^2 + c^2 + d^2 = 2018.$

2010 IMO Shortlist, 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]

2000 ITAMO, 1

Tags: number theory , consecutive , sum of squares

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?

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 $n$.

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.

2024 Austrian MO National Competition, 6

Tags: modular arithmetic , modulo , sum of squares , number theory

For each prime number $p$, determine the number of residue classes modulo $p$ which can be represented as $a^2+b^2$ modulo $p$, where $a$ and $b$ are arbitrary integers. [i](Daniel Holmes)[/i]

2010 Contests, 1

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

2010 Saudi Arabia Pre-TST, 2.2

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

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

1991 Tournament Of Towns, (281) 1

Tags: number theory , sum of squares

$N$ integers are given. Prove that the sum of their squares is divisible by $N$ if it is known that the difference between the product of any $N - 1$ of them and the last one is divisible by $N$. (D. Fomin, Leningrad)

1978 IMO Longlists, 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.

2010 Greece JBMO TST, 1

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

2016 Polish MO Finals, 5

Tags: algebra , real number , sum of squares

There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

2014 Federal Competition For Advanced Students, P2, 4

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

For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?

1977 Spain Mathematical Olympiad, 4

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

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

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.

2018 Romania National Olympiad, 1

Tags: number theory , perfect square , sum of squares

Prove that there are infinitely many sets of four positive integers so that the sum of the squares of any three elements is a perfect square.

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)

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 $0$?

2015 Balkan MO Shortlist, N1

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

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

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 $3$.

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