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

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.

2025 Macedonian Balkan MO TST, 4
Tags: sum of squares , prime , number theory
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 $p = a^2 + 7b^2$.

1989 All Soviet Union Mathematical Olympiad, 494
Tags: sum of squares , 5-digit , subset , algebra , number theory
Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.

2015 Bosnia Herzegovina Team Selection Test, 4
Tags: sum of squares , set , combinatorics
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 $B$.

2014 India PRMO, 5
Tags: algebra , sum of squares
If real numbers $a, b, c, d, e$ satisfy $a + 1 = b + 2 = c + 3 = d + 4 = e + 5 = a + b + c + d + e + 3$, what is the value of $a^2 + b^2 + c^2 + d^2 + e^2$ ?

2016 India Regional Mathematical Olympiad, 5
Tags: sum of squares , number theory
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.

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?

2015 Balkan MO Shortlist, N1
Tags: number theory , sum of squares , square , combinatorics
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)

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.

2024 Austrian MO National Competition, 6
Tags: modulo , sum of squares , modular arithmetic , 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]

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: $1$. $a< \frac{p}{q} < \frac{r}{s} < b$. $2.$ $p^2+q^2=r^2+s^2$.

2022 JBMO Shortlist, N6
Tags: sum of squares , digit , shortlist , number theory
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 $n$.

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.

2019 Brazil Team Selection Test, 4
Tags: set , prime number , sum of squares , number theory
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.

1991 Tournament Of Towns, (281) 1
Tags: sum of squares , number theory
$N$ 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)

1949-56 Chisinau City MO, 8
Tags: remainder , sum of squares , number theory
Prove that the remainder of dividing the sum of two squares of integers by $4$ is different from $3$.

2016 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: game strategy , sum of squares , game , combinatorics
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer $N$ and announces it to Beto. Next Beto writes a succession of $2016$ numbers, $1008$ of them equal to $1$ and $1008$ of them equal to $-1$. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to $N$, Ana wins. If not, Beto wins. Determine all values of $N$ for which Ana can ensure victory, no matter how Beto plays.

2019 Tournament Of Towns, 4
Tags: consecutive , multiple , sum of squares , circles , number theory
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)

1952 Moscow Mathematical Olympiad, 216
Tags: number theory , integer sequence , sequence , sum of squares , digit
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 $a_1, a_2, a_3, ....$

2014 Federal Competition For Advanced Students, P2, 4
Tags: consecutive , sum of squares , product , divisor , number theory
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 $P (n)$ ?

2018 Romania National Olympiad, 1
Tags: sum of squares , number theory
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) $a^2 + b^2 + c^2 + d^2 = 2018.$

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 $3$.

2017 Thailand Mathematical Olympiad, 5
Tags: perfect square , sum of squares , consecutive , number theory
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.

2017 Regional Competition For Advanced Students, 4
Tags: sum of squares , divisibility , divisor , number theory
Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

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: $$n = \pm 1^2 \pm 2^2 \pm 3^2 \pm ... \pm k^2$$