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: 521

2000 Abels Math Contest (Norwegian MO), 1b
Tags: perfect square , number theory
Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square

1996 IMO Shortlist, 2
Tags: modular arithmetic , number theory , quadratic reciprocity , perfect square
The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2020 Dutch Mathematical Olympiad, 4
Tags: number theory , perfect square , prime
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , polynomial , floor function , perfect square
Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. [i]Proposed by Ilir Snopce[/i]

2014 Finnish National High School Mathematics, 5
Tags: minimum , sum , number theory , combinatorics , perfect square
Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$. In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?

2002 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , perfect square , digit
The last four digits of a perfect square are equal. Prove that all of them are zeros.

1994 Argentina National Olympiad, 2
Tags: number theory , perfect square , algebra
For what positive integer values of $x$ is $x^4 + 6x^3 + 11x^2 + 3x + 31$ a perfect square?

2016 Cono Sur Olympiad, 1
Tags: number theory , perfect square
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers. [b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.

1996 Tournament Of Towns, (496) 3
Tags: factorial , number theory , perfect square
Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square? (S Tokarev)

1994 IMO Shortlist, 1
Tags: number theory , extremal combinatorics , subset , perfect square
$ M$ is a subset of $ \{1, 2, 3, \ldots, 15\}$ such that the product of any three distinct elements of $ M$ is not a square. Determine the maximum number of elements in $ M.$

2019 Swedish Mathematical Competition, 6
Tags: sequence , perfect square , number theory
Is there an infinite sequence of positive integers $\{a_n\}_{n = 1}^{\infty}$ which contains each positive integer exactly once and is such that the number $a_n + a_{n + 1} $ is a perfect square for each $n$?

2019 Regional Olympiad of Mexico West, 3
Tags: number theory , perfect square
Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$ is the square of a prime number.

1965 Polish MO Finals, 4
Tags: number theory , perfect square
Prove that if the integers $ a $ and $ b $ satisfy the equation $$ 2a^2 + a = 3b^2 + b,$$ then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.

1997 German National Olympiad, 1
Tags: perfect square , number theory
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.

2022 Pan-African, 2
Tags: number theory , perfect square , algebra
Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.

2020 Junior Balkan Team Selection Tests - Moldova, 10
Tags: number theory , prime number , perfect square
Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.

2004 Estonia National Olympiad, 3
Tags: number theory , digit , perfect square
The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.

2021 Belarusian National Olympiad, 10.5
Tags: number theory , perfect square
Prove that for any positive integer $n$ there exist infinitely many triples $(a,b,c)$ of pairwise distinct positive integers such that $ab+n,bc+n,ac+n$ are all perfect squares

2012 Czech-Polish-Slovak Junior Match, 5
Tags: number theory , pythagorean triple , perfect square
Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$. Show that the number $\frac12(c - a) (c - b)$ is square of an integer.

2019 Durer Math Competition Finals, 3
Tags: perfect square , number theory
Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.

1991 Bundeswettbewerb Mathematik, 1
Tags: number theory , diophantine equation , diophantine , perfect square
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P5
Tags: algebra , binary , perfect square , memorable
We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares? [i]Proposed by Nikola Velov[/i]

2021 Malaysia IMONST 1, 15
Tags: number theory , perfect square
Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.

2009 Belarus Team Selection Test, 1
Tags: perfect square , number theory
Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares. S. Kuzmich

KoMaL A Problems 2017/2018, A. 717
Tags: number theory , prime number , perfect square
Let's call a positive integer $n$ special, if there exist two nonnegativ integers ($a, b$), such that $n=2^a\times 3^b$. Prove that if $k$ is a positive integer, then there are at most two special numbers greater then $k^2$ and less than $k^2+2k+1$.