Found problems: 521
2018 Greece Junior Math Olympiad, 3
Let $a$ and $b$ be positive integers with $b$ odd, such that the number $$\frac{(a+b)^2+4a}{ab}$$ is an integer. Prove that $a$ is a perfect square.
Fractal Edition 2, P1
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 $\frac{a+b}{b+c}$.
2021 European Mathematical Cup, 3
Let $\ell$ be a positive integer. We say that a positive integer $k$ is [i]nice [/i] if $k!+\ell$ is a square of an integer. Prove that for every positive integer $n \geqslant \ell$, the set $\{1, 2, \ldots,n^2\}$ contains at most $n^2-n +\ell$ nice integers. \\ \\
(Théo Lenoir)
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2008 Tournament Of Towns, 1
An integer $N$ is the product of two consecutive integers.
(a) Prove that we can add two digits to the right of this number and obtain a perfect square.
(b) Prove that this can be done in only one way if $N > 12$
1998 Romania Team Selection Test, 2
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.
2020 Final Mathematical Cup, 1
Let $n$ be a given positive integer. Prove that there is no positive divisor $d$ of $2n^2$ such that $d^2n^2+d^3$ is a square of an integer.
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.
2023 239 Open Mathematical Olympiad, 4
We call a natural number [i]almost a square[/i] if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.
2021 Abels Math Contest (Norwegian MO) Final, 3a
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?
1963 Swedish Mathematical Competition., 1
How many positive integers have square less than $10^7$?
1964 Dutch Mathematical Olympiad, 3
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.
2014 Hanoi Open Mathematics Competitions, 4
Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square.
(A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above.
2005 iTest, 34
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
1992 Tournament Of Towns, (352) 1
Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square.
(S Tokarev)
2020 Regional Olympiad of Mexico Northeast, 4
Let \(n > 1\) be an integer and \(p\) be a prime. Prove that if \(n|p-1\) and \(p|n^3-1\), then \(4p-3\) is 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 $S(n)=n^{2}-14.$
OIFMAT III 2013, 1
Find all four-digit perfect squares such that:
$\bullet$ All your figures are less than $9$.
$\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.
2021 IMO Shortlist, N2
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.
2019 India PRMO, 14
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
1996 Denmark MO - Mohr Contest, 4
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
1988 IMO Shortlist, 9
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.
2016 District Olympiad, 2
If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $
1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.
2017 Rioplatense Mathematical Olympiad, Level 3, 4
Is there a number $n$ such that one can write $n$ as the sum of $2017$ perfect squares and (with at least) $2017$ distinct ways?
2022 Pan-African, 2
Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.