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.

AND:
OR:
NO:

Found problems: 521

2023 Czech-Polish-Slovak Junior Match, 1
Tags: number theory , perfect square , sum of digits
Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

2010 IMO, 3
Tags: function , perfect square , number theory , algebra
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

2012 Czech-Polish-Slovak Junior Match, 3
Tags: perfect square , number theory
Prove that if $n$ is a positive integer then $2 (n^2 + 1) - n$ is not a square of an integer.

2008 IMAC Arhimede, 1
Tags: perfect square , prime , number theory
Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

2011 Bundeswettbewerb Mathematik, 2
Tags: prime , composite , composite number , perfect square , number theory
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.

2010 Dutch IMO TST, 3
Tags: perfect square , number theory
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

1984 All Soviet Union Mathematical Olympiad, 387
Tags: perfect square , number theory
The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.

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.

2019 Regional Olympiad of Mexico Center Zone, 1
Tags: number theory , perfect square
Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.

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

1994 IMO Shortlist, 1
Tags: subset , perfect square , extremal combinatorics , number theory
$ 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.$

2008 Brazil Team Selection Test, 4
Tags: perfect square , phi function , euler s phi function , number theory
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2011 Vietnam Team Selection Test, 5
Tags: modular arithmetic , perfect square , number theory , induction
Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.

2002 Portugal MO, 5
Tags: perfect square , square , geometry
Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

2022 Durer Math Competition Finals, 4
Tags: perfect square , divisor , number theory
Show that the divisors of a number $n \ge 2$ can only be divided into two groups in which the product of the numbers is the same if the product of the divisors of $n$ is a square number.

1989 Tournament Of Towns, (235) 3
Tags: perfect square , number theory , combinatorics
Do there exist $1000 000$ distinct positive integers such that the sum of any collection of these numbers is never an exact square?

2022 Canadian Mathematical Olympiad Qualification, 2
Tags: perfect square , number theory
Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

2008 Thailand Mathematical Olympiad, 8
Tags: perfect square , number theory
Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

2006 Belarusian National Olympiad, 8
Tags: perfect square , number theory
a) Do there exist positive integers $a$ and $b$ such that for any positive,integer $n$ the number $a \cdot 2^n+ b\cdot 5^n$ is a perfect square ? b) Do there exist positive integers $a, b$ and $c$, such that for any positive integer $n$ the number $a\cdot 2^n+ b\cdot 5^n + c$ is a perfect square? (M . Blotski)

1941 Moscow Mathematical Olympiad, 075
Tags: number theory , perfect square , consecutive
Prove that $1$ plus the product of any four consecutive integers is a perfect square.

2013 IMO Shortlist, N4
Tags: perfect square , decimal representation , number theory
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

2009 Abels Math Contest (Norwegian MO) Final, 1a
Tags: perfect square , number theory
Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

1984 IMO Shortlist, 10
Tags: equation , number theory , product , perfect square
Prove that the product of five consecutive positive integers cannot be the square of an integer.

1999 Switzerland Team Selection Test, 6
Tags: divisible , perfect square , number theory
Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

2018 Junior Balkan Team Selection Tests - Romania, 1
Tags: perfect square , number theory
Determine the prime numbers $p$ for which the number $a = 7^p - p - 16$ is a perfect square. Lucian Petrescu