Found problems: 521
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]
1989 Swedish Mathematical Competition, 1
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.
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.
2012 Estonia Team Selection Test, 1
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.
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.
1999 All-Russian Olympiad Regional Round, 10.5
Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?
2018 Bosnia And Herzegovina - Regional Olympiad, 3
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
2019 Saudi Arabia Pre-TST + Training Tests, 2.1
Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.
Russian TST 2015, P3
Find all integers $k{}$ for which there are infinitely many triples of integers $(a,b,c)$ such that \[(a^2-k)(b^2-k)=c^2-k.\]
2000 Belarus Team Selection Test, 6.2
A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$.
Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.
2012 Cuba MO, 8
If the natural numbers $a, b, c, d$ verify the relationships:
$$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$
$$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$
and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
1992 IMO, 3
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 $S(n)=n^{2}-14.$
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?
1951 Moscow Mathematical Olympiad, 199
Prove that the sum $1^3 + 2^3 +...+ n^3$ is a perfect square for all $n$.
2019 Durer Math Competition Finals, 3
Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
2022 Romania Team Selection Test, 4
Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?
1998 Belarus Team Selection Test, 3
a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer.
b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?
1991 Bundeswettbewerb Mathematik, 1
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.
2024 Mexican University Math Olympiad, 1
Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.
2011 District Olympiad, 2
a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$.
b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$
1936 Moscow Mathematical Olympiad, 022
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
1998 Romania National Olympiad, 1
Let $n$ be a positive integer and $x_1,x_2,...,x_n$ be integer numbers such that $$x_1^2+x_2^2+...+x_n^2+ n^3 \le (2n - 1)(x_1+x_2+...+x_n ) + n^2$$ .
Show that :
a) $x_1,x_2,...,x_n$ are non-negative integers
b) the number $x_1+x_2+...+x_n+n+1$ is not a perfect square.
2006 Chile National Olympiad, 4
Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .