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

2013 Hanoi Open Mathematics Competitions, 1
Tags: number theory , perfect square
How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square? (A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.

2010 IMO, 3
Tags: perfect square , function , algebra , number theory
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]

2022 Grand Duchy of Lithuania, 4
Tags: number theory , perfect square
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

2004 Switzerland Team Selection Test, 2
Tags: perfect square , number theory
Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

1991 ITAMO, 2
Tags: perfect square , number theory
Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

2017 Saudi Arabia JBMO TST, 6
Tags: number theory , perfect square
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.

1926 Eotvos Mathematical Competition, 2
Tags: number theory , perfect square , consecutive
Prove that the product of four consecutive natural numbers cannot be the square of an integer.

2023 AMC 10, 15
Tags: perfect square , number theory
What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square? $\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

2015 Peru MO (ONEM), 3
Tags: perfect square , number theory , floor function
Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

1963 All Russian Mathematical Olympiad, 036
Tags: number theory , perfect square
Given the endless arithmetic progression with the positive integer members. One of those is an exact square. Prove that the progression contain the infinite number of the exact squares.

2005 Argentina National Olympiad, 4
Tags: number theory , perfect cube , perfect square , geometry , 3d geometry
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

2008 Tournament Of Towns, 1
Tags: number theory , consecutive , perfect square
An integer $N$ is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if $N > 12$

2019 Olympic Revenge, 2
Tags: perfect square , number theory , prime number , my 100th post
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$

1999 Spain Mathematical Olympiad, 2
Tags: number theory , perfect square , integer sequence
Prove that there exists a sequence of positive integers $a_1,a_2,a_3, ...$ such that $a_1^2+a_2^2+...+a_n^2$ is a perfect square for all positive integers $n$.

2018 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , perfect square
Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.

2020 Iran MO (2nd Round), P5
Tags: perfect square , number theory
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.

1997 Tuymaada Olympiad, 1
Tags: number theory , product , perfect square
The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

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

2015 Silk Road, 2
Tags: perfect square , arithmetic progression , sequence , number theory
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .

1974 Dutch Mathematical Olympiad, 4
Tags: perfect square , number theory
For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

2015 Thailand Mathematical Olympiad, 8
Tags: perfect square , odd , number theory
Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

2020 Greece JBMO TST, 3
Tags: perfect square , number theory
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.

2014 IFYM, Sozopol, 2
Tags: number theory , perfect square
Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?

2012 Argentina National Olympiad, 2
Tags: number theory , perfect square
Determine all natural numbers $n$ for which there are $2n$ distinct positive integers $x_1,…,x_n,y_1,…,y_n$ such that the product $$(11x^2_1+12y^2_1)(11x^2_2+12y^2_2)…(11x^2_n+12y^2_n)$$ is a perfect square.

1981 Swedish Mathematical Competition, 1
Tags: perfect square , number theory
Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.