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

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 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.

2014 IFYM, Sozopol, 3
Tags: algebra , polynomial , number theory , Sum of Squares , IMO Shortlist
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]

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 $3$.

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

1952 Moscow Mathematical Olympiad, 216
Tags: Integer sequence , Sequence , Sum of Squares , Digits , 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 $a_1, a_2, a_3, ....$

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 $13^k$.

2016 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: game strategy , combinatorics , Sum of Squares , game
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.

1999 Tournament Of Towns, 3
Tags: number theory , divides , 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)

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.

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 $0$?

1985 All Soviet Union Mathematical Olympiad, 396
Tags: sum of digits , Sum of Squares , Sum , number theory , decimal representation , Digits
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 $1000000$?

2015 Thailand TSTST, 1
Tags: number theory , square , Perfect Squares , 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.

2017 Thailand Mathematical Olympiad, 5
Tags: number theory , Perfect Squares , Sum of Squares , consecutive
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.

2021 Poland - Second Round, 3
Tags: number theory , Sum of Squares
Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

1989 All Soviet Union Mathematical Olympiad, 494
Tags: Sum of Squares , Subset , number theory , 5-digit , algebra
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.

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.

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)

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

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 $3$.

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 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?

2022 JBMO Shortlist, N6
Tags: number theory , Sum of Squares , Digits , Junior , Balkan , 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 $n$.