Found problems: 521
2021 Argentina National Olympiad, 5
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.
2009 All-Russian Olympiad Regional Round, 9.2
Rational numbers $a$ and $b$ satisfy the equality $$a^3b+ab^3+2a^2b^2+2a + 2b + 1 = 0. $$ Prove that the number $1-ab$ is the square of the rational numbers.
1988 IMO Shortlist, 25
A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.
2018 Ukraine Team Selection Test, 7
The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number.
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2022 Bulgaria National Olympiad, 3
Let $x>y>2022$ be positive integers such that $xy+x+y$ is a perfect square. Is it possible for every positive integer $z$ from the interval $[x+3y+1,3x+y+1]$ the numbers $x+y+z$ and $x^2+xy+y^2$ not to be coprime?
2003 Junior Balkan Team Selection Tests - Moldova, 1
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.
2006 All-Russian Olympiad Regional Round, 9.8
A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.
India EGMO 2021 TST, 6
Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$.
Show that $a^2+b^2-ab$ is not a square.
ICMC 6, 4
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
1976 Swedish Mathematical Competition, 3
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.
2000 Singapore Team Selection Test, 2
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square
2015 Silk Road, 2
Let $\left\{ {{a}_{n}} \right\}_{n \geq 1}$ and $\left\{ {{b}_{n}} \right\}_{n \geq 1}$ be two infinite arithmetic progressions, each of which the first term and the difference are mutually prime natural numbers. It is known that for any natural $n$, at least one of the numbers $\left( a_n^2+a_{n+1}^2 \right)\left( b_n^2+b_{n+1}^2 \right) $ or $\left( a_n^2+b_n^2 \right) \left( a_{n+1}^2+b_{n+1}^2 \right)$ is an perfect square. Prove that ${{a}_{n}}={{b}_{n}}$, for any natural $n$ .
1931 Eotvos Mathematical Competition, 2
Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.
2018 Ukraine Team Selection Test, 11
$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition [i]honest[/i], if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.
2007 Thailand Mathematical Olympiad, 4
Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.
2000 Abels Math Contest (Norwegian MO), 1b
Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square
1992 IMO Longlists, 22
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 $S(n)=n^{2}-14.$
1969 Kurschak Competition, 1
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).
2021 Argentina National Olympiad, 2
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$
2014 IFYM, Sozopol, 2
Does there exist a natural number $n$, for which $n.2^{2^{2014}}-81-n$ is a perfect square?
2015 Abels Math Contest (Norwegian MO) Final, 4
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
II Soros Olympiad 1995 - 96 (Russia), 10.5
Find all pairs of natural numbers $x$ and $y$ for which $x^2+3y$ and $y^2+3x$ are simultaneously squares of natural numbers.
2013 Dutch IMO TST, 2
Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.
1964 All Russian Mathematical Olympiad, 054
Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.
2020 Romanian Master of Mathematics Shortlist, N2
For a positive integer $n$, let $\varphi(n)$ and $d(n)$ denote the value of the Euler phi function at $n$ and the number of positive divisors of $n$, respectively. Prove that there are infinitely many positive integers $n$ such that $\varphi(n)$ and $d(n)$ are both perfect squares.
[i]Finland, Olli Järviniemi[/i]