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

2015 India PRMO, 6
Tags: perfect square , number theory , digit , sum of digits
$6.$ How many two digit positive integers $N$ have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a perfect square $?$

2014 Denmark MO - Mohr Contest, 4
Tags: perfect square , number theory
Determine all positive integers $n$ so that both $20n$ and $5n + 275$ are perfect squares. (A perfect square is a number which can be expressed as $k^2$, where $k$ is an integer.)

2024 Tuymaada Olympiad, 3
Tags: algebra , perfect square , sequence
All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers \[f(n),f(2n),f(3n),\dots,f(10n^2)\] are perfect squares?

2010 Saudi Arabia BMO TST, 1
Tags: perfect square , number theory
Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.

2009 Thailand Mathematical Olympiad, 1
Tags: number theory , perfect square
Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

2018 Saudi Arabia IMO TST, 2
Tags: combinatorics , perfect square , number theory
Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

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.

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.

2019 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , divisor , perfect square
Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

2013 Hanoi Open Mathematics Competitions, 2
Tags: number theory , perfect square
How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

2021 Regional Olympiad of Mexico West, 2
Tags: number theory , perfect square
Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

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.

2012 Estonia Team Selection Test, 1
Tags: perfect square , perfect cube , number theory
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

1936 Moscow Mathematical Olympiad, 022
Tags: perfect square , 4-digit , number theory
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.

2017 CHMMC (Fall), 5
Tags: perfect square , number theory
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

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 \[x^2+y^2=3z^2.\]

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.

2023 Czech-Polish-Slovak Junior Match, 1
Tags: number theory , perfect square , sum of digits
Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

2010 Dutch BxMO TST, 5
Tags: quadratic , permutation , 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 $\{0,1,..., n\}$.

1976 Swedish Mathematical Competition, 3
Tags: algebra , perfect square , rational
If $a$, $b$, $c$ are rational, show that \[ \frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2} \] is the square of a rational.

Fractal Edition 1, P3
Tags: perfect square
Can the number \( \overline{abc} + \overline{bca} + \overline{cab} \) be a perfect square?

2012 Flanders Math Olympiad, 2
Tags: perfect square , number theory
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.

2019 New Zealand MO, 4
Tags: number theory , perfect square
Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

2001 Polish MO Finals, 1
Tags: modular arithmetic , function , number theory , perfect square
Assume that $a,b$ are integers and $n$ is a natural number. $2^na+b$ is a perfect square for every $n$.Prove that $a=0$.

1987 Greece National Olympiad, 2
Tags: number theory , perfect square
Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .