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

1997 IMO Shortlist, 15
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.

2008 Postal Coaching, 2
Tags: polynomial , integer polynomial , perfect square , algebra
Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

2000 Romania National Olympiad, 1
Tags: number theory , perfect square
a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

2022 AMC 12/AHSME, 16
Tags: perfect square
A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? $\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $

2007 Estonia Math Open Junior Contests, 10
Tags: combinatorics , sum , number theory , perfect square
Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.

2011 Denmark MO - Mohr Contest, 1
Tags: perfect square , combinatorics , number theory
Georg writes the numbers from $1$ to $15$ on different pieces of paper. He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible. (The square numbers are the numbers $0 = 0^2$, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$ etc.)

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]

1986 All Soviet Union Mathematical Olympiad, 423
Tags: perfect square , table , number theory , sum , combinatorics
Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.

2013 AIME Problems, 6
Tags: perfect square , modular arithmetic , number theory
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.

OIFMAT III 2013, 4
Tags: perfect power , perfect square , 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?

2019 Saudi Arabia JBMO TST, 4
Tags: number theory , perfect square
Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

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.

2024 Brazil EGMO TST, 4
Tags: perfect square , number theory , sequence
The infinite sequence \( a_1, a_2, \ldots \) is defined by \( a_1 = 1 \) and, for each \( n \geq 1 \), the number \( a_{n+1} \) is the smallest positive integer greater than \( a_n \) that has the following property: for each \( k \in \{1, 2, \ldots, n\} \), the number \( a_{n+1} + a_k \) is not a perfect square. Prove that, for all \( n \), it holds that \( a_n \leq (n - 1)^2 + 1 \).

2002 Brazil National Olympiad, 1
Tags: perfect square , number theory , perfect cube , perfect power , sum , subset
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

Fractal Edition 2, P1
Tags: perfect square
The positive integers $a$, $b$, $c$ are such that $\frac{a+b}{b+c}$ is the square of a rational number, and $ab+bc+ca$ is a prime number. Find all possible values of $\frac{a+b}{b+c}$.

1988 Austrian-Polish Competition, 5
Tags: perfect square , number theory , recurrence relation , sequence
Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.

2006 Thailand Mathematical Olympiad, 1
Tags: number theory , consecutive , perfect square
Show that the product of three consecutive positive integers is never a perfect square.

2009 Silk Road, 4
Tags: number theory , perfect square , prime
Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.

2018 Polish Junior MO Second Round, 5
Tags: combinatorics , coloring , number theory , perfect square
Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.

2013 Nordic, 1
Tags: floor function , perfect square , number theory , integer sequence
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$, where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$. Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square

2019 Saudi Arabia Pre-TST + Training Tests, 3.3
Tags: perfect square , product , recurrence relation , number theory , sequence
Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.

1988 IMO Longlists, 14
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.

2014 Belarus Team Selection Test, 3
Tags: number theory , perfect square , decimal representation
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.

1994 Spain Mathematical Olympiad, 1
Tags: arithmetic progression , number theory , perfect square
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

2010 May Olympiad, 3
Tags: number theory , consecutive , perfect square
Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.