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

1988 IMO, 3
Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

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.

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 $y$.

1998 Akdeniz University MO, 1
Tags: number theory , equation , perfect square
Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$ is not a perfect square.

1998 Romania Team Selection Test, 2
Tags: number theory , modular arithmetic , perfect square , perfect cube , arithmetic sequence
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

1979 IMO Shortlist, 23
Tags: number theory , perfect square
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

2021 Brazil National Olympiad, 3
Tags: floor function , irrational number , perfect square , power , number theory
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).

2018 India PRMO, 15
Tags: minimum , number theory , perfect square
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?

2024 JBMO TST - Turkey, 5
Tags: number theory , divisibility , perfect square
Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.

1990 Bundeswettbewerb Mathematik, 2
Tags: sequence , perfect square , recurrence relation , algebra
The sequence $a_0,a_1,a_2,...$ is defined by $a_0 = 0, a_1 = a_2 = 1$ and $a_{n+2} +a_{n-1} = 2(a_{n+1} +a_n)$ for all $n \in N$. Show that all $a_n$ are perfect 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.

OIFMAT III 2013, 1
Tags: number theory , digit , perfect square
Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

1969 IMO Shortlist, 62
Tags: number theory , perfect square
Which natural numbers can be expressed as the difference of squares of two integers?

1917 Eotvos Mathematical Competition, 2
Tags: number theory , perfect square
In the square of an integer $ a$, the tens’ digit is $7$. What is the units’ digit of $a^2$?

OIFMAT III 2013, 4
Tags: perfect square , perfect power , number theory
Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

2021 Olympic Revenge, 5
Tags: number theory , diophantine equation , arithmetic sequence , perfect square
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.

1993 Bundeswettbewerb Mathematik, 3
Tags: number theory , perfect square
There are pairs of square numbers with the following two properties: (1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ . (2) If one appends the second to the decimal representation of the first, the decimal representation results another square number. Example: $16$ and $81$; $1681 = 41^2$. Prove that there are infinitely many pairs of squares with these properties.

2005 Bosnia and Herzegovina Junior BMO TST, 2
Tags: perfect square , integer , number theory
Let n be a positive integer. Prove the following statement: ”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”

2023 Stars of Mathematics, 2
Tags: number theory , perfect square
Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$. [i]Proposed by Vlad Matei[/i]

2015 Thailand TSTST, 1
Tags: number theory , square , sum of squares , perfect square
Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

2001 Romania Team Selection Test, 4
Tags: number theory , additive number theory , sum of squares , perfect square
Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.