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

2018 Romania National Olympiad, 1
Tags: number theory , Perfect Squares , 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 China Western Mathematical Olympiad, 1
Tags: number theory , Perfect Square
Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.

1989 Swedish Mathematical Competition, 1
Tags: number theory , Digits , Perfect Square
Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.

2019 IMO Shortlist, N8
Tags: ceiling function , number theory , IMO Shortlist , IMO Shortlist 2019 , Perfect Square
Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

1997 All-Russian Olympiad Regional Round, 8.1
Tags: number theory , Perfect Square , combinatorics
Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.

2010 Thailand Mathematical Olympiad, 1
Tags: number theory , Perfect Square
Show that, for every positive integer $x$, there is a positive integer $y\in \{2, 5, 13\}$ such that $xy - 1$ is not a perfect square.

1994 All-Russian Olympiad Regional Round, 9.7
Tags: number theory , Perfect Square
Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.

2018 Saudi Arabia IMO TST, 3
Tags: Perfect Square , number theory
Two sets of positive integers $A, B$ are called [i]connected [/i] if they are not empty and for all $a \in A, b \in B$, number $ab + 1$ is a perfect square. i) Given $A =\{1, 2,3, 4\}$. Prove that there does not exist any set $B$ such that $A, B$ are connected. ii) Suppose that $A, B$ are connected with $|A|,|B| \ge 2$. For any $a_1 > a_2 \in A$ and $b_1 > b_2 \in B$, prove that $a_1b_1 > 13a_2b_2$.

2019 Regional Olympiad of Mexico West, 3
Tags: number theory , Perfect Square
Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$ is the square of a prime number.

2008 Flanders Math Olympiad, 2
Tags: number theory , Perfect Square
Let $a, b$ and $c$ be integers such that $a+b+c = 0$. Prove that $\frac12(a^4 +b^4 +c^4)$ is a perfect square.

2021 Abels Math Contest (Norwegian MO) Final, 3a
Tags: number theory , Perfect Squares , Perfect Square
For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ is a perfect square?

1988 Bundeswettbewerb Mathematik, 1
Tags: number theory , Perfect Squares , Perfect Square
For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ . Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.

1977 Spain Mathematical Olympiad, 4
Tags: Perfect Square , Sum of Squares , consecutive , number theory
Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

1997 Singapore Senior Math Olympiad, 3
Tags: number theory , Digits , Perfect Square
Find the smallest positive integer $x$ such that $x^2$ ends with the four digits $9009$.

1965 Polish MO Finals, 4
Tags: number theory , Perfect Square
Prove that if the integers $ a $ and $ b $ satisfy the equation $$ 2a^2 + a = 3b^2 + b,$$ then the numbers $ a - b $ and $ 2a + 2b + 1 $ are squares of integers.

1999 Switzerland Team Selection Test, 10
Tags: Product , consecutive , Perfect Square , number theory
Prove that the product of five consecutive positive integers cannot be a perfect square.

2019 Tournament Of Towns, 5
Tags: Perfect Squares , Perfect Square , number theory
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice. (Yury Markelov)

2003 Junior Balkan Team Selection Tests - Moldova, 1
Tags: Sum , number theory , Perfect Square
Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square. Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.

2010 Dutch IMO TST, 3
Tags: number theory , Perfect Square
Let $n\ge  2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2015 Abels Math Contest (Norwegian MO) Final, 4
Tags: number theory , exponential , Perfect Square
a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even. b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd

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

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.

2019 Grand Duchy of Lithuania, 4
Tags: number theory , Perfect Square
Determine all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a square of an integer.

1976 Czech and Slovak Olympiad III A, 1
Tags: number theory , Diophantine equation , Perfect Square , Sum of Squares
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]

2005 iTest, 27
Tags: number theory , Perfect Square , Digits
Find the sum of all non-zero digits that can repeat at the end of a perfect square. (For example, if $811$ were a perfect square, $1$ would be one of these non-zero digits.)