Found problems: 171
2022 Czech and Slovak Olympiad III A, 5
Find all integers $n$ such that $2^n + n^2$ is a square of an integer.
[i](Tomas Jurik )[/i]
1986 IMO, 1
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
1992 IMO Shortlist, 21
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
[b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
[b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$.
[b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$
2023 Stars of Mathematics, 2
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]
2024 Brazil EGMO TST, 4
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 \( a_n \leq (n - 1)^2 + 1 \).
2011 Bundeswettbewerb Mathematik, 2
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.
1988 Greece National Olympiad, 4
Prove that there are do not exist natural numbers $k, m$ such that numbers $k^2+2m$, $m^2+2k$ to be squares of integers.
2021 Olympic Revenge, 5
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
2020 Peru Iberoamerican Team Selection Test, P4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2020 EGMO, 6
Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$, $a_3 = 4$, and for all $n \ge 4$, $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$
Determine all integers $m$ such that every term of the sequence is a square.
2001 China Team Selection Test, 1
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
2019 Olympic Revenge, 2
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$
2024 Turkey Team Selection Test, 4
Find all positive integer pairs $(a,b)$ such that, $$\frac{10^{a!} - 3^b +1}{2^a}$$ is a perfect square.
2021 Azerbaijan Senior NMO, 2
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.
2025 Israel National Olympiad (Gillis), P3
Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.
2009 Regional Olympiad of Mexico Center Zone, 4
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.
2017 Czech-Polish-Slovak Match, 1
Find all positive real numbers $c$ such that there are in finitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer.
(Slovakia)
2024 Bundeswettbewerb Mathematik, 2
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?
2021 Belarusian National Olympiad, 10.5
Prove that for any positive integer $n$ there exist infinitely many triples $(a,b,c)$ of pairwise distinct positive integers such that $ab+n,bc+n,ac+n$ are all perfect squares
2004 India IMO Training Camp, 3
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
[i]Proposed by Hojoo Lee, Korea[/i]
2020 Junior Balkan Team Selection Tests - Moldova, 10
Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.
2018 Romania National Olympiad, 1
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.
2024 Junior Macedonian Mathematical Olympiad, 4
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
2020 Peru Cono Sur TST., P4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2022 AMC 12/AHSME, 16
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 $