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: 397

2016 Bosnia and Herzegovina Junior BMO TST, 1
Tags: number theory , Perfect Square
Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

2021 IMO Shortlist, N2
Tags: IMO , Perfect Square , algebra , combinatorics , number theory
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

2006 All-Russian Olympiad Regional Round, 11.7
Tags: Perfect Square , Sum of Squares , number theory
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.

2021 Azerbaijan Senior NMO, 2
Tags: number theory , Perfect Squares , 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.

1999 Austrian-Polish Competition, 5
Tags: recurrence relation , Sequence , Perfect Square , number theory
A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$ Prove that there exists at most one $n$ for which $a_n$ is a perfect square.

1952 Moscow Mathematical Olympiad, 224
Tags: number theory , Perfect Square
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,. b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places

2017 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: Perfect Square , arranging , combinatorics , number theory
Prove that numbers $1,2,...,16$ can be divided in sequence such that sum of any two neighboring numbers is perfect square

1986 Poland - Second Round, 4
Tags: number theory , Perfect Square , diophantine , Diophantine equation
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$ Prove that $ x + y $ is the square of a natural number.

2004 Mexico National Olympiad, 1
Tags: primes , Perfect Square , number theory
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

2017 QEDMO 15th, 3
Tags: number theory , Perfect Square
Let $a,b,c$ natural numbers for which $a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2$. Show that $ab, bc, ca$ and $ab + bc + ca$ are perfect squares .

2010 Chile National Olympiad, 1
Tags: number theory , Perfect Square
The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.

2003 Croatia Team Selection Test, 1
Tags: Perfect Square , number theory
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

2010 Dutch BxMO TST, 5
Tags: permutation , Quadratic , number theory , Perfect Square
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

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

2021 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , Perfect Square
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.

2020-21 IOQM India, 8
Tags: number theory , Perfect Square , Digit
A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

2021 IMO, 1
Tags: IMO , Perfect Square , algebra , combinatorics , number theory
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

2009 Regional Olympiad of Mexico Center Zone, 4
Tags: Perfect Square , Perfect Squares , consecutive , Digits , number theory
Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

1998 Junior Balkan MO, 1
Tags: induction , Perfect Square
Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of 1-s and 1998 of 2-s) is a perfect square.

1998 Romania National Olympiad, 2
Tags: number theory , Perfect Square
Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.

2024 Bundeswettbewerb Mathematik, 2
Tags: number theory , number theory proposed , Digits , Perfect Squares , Perfect Square
Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

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.

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 $n$?

2013 NZMOC Camp Selection Problems, 8
Tags: Perfect Square , number theory
Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

2013 Czech-Polish-Slovak Match, 1
Tags: Perfect Square , trinomial , number theory , Integer Polynomial
Let $a$ and $b$ be integers, where $b$ is not a perfect square. Prove that $x^2 + ax + b$ may be the square of an integer only for finite number of integer values of $x$. (Martin Panák)