Found problems: 521
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]
1996 Dutch Mathematical Olympiad, 2
Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.
2000 Abels Math Contest (Norwegian MO), 1a
Show that any odd number can be written as the difference between two perfect squares.
2010 Dutch BxMO TST, 5
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 $\{0,1,..., n\}$.
2020 Thailand TST, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
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.
1992 Tournament Of Towns, (333) 1
Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer.
(V. Senderov, Moscow)
2011 Bosnia And Herzegovina - Regional Olympiad, 4
For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square
2008 Flanders Math Olympiad, 2
Let $a, b$ and $c$ be integers such that $a+b+c = 0$. Prove that $\frac12(a^4 +b^4 +c^4)$ is a perfect square.
2003 Croatia Team Selection Test, 1
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
2023 Peru MO (ONEM), 1
We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$.
a) What is the smallest positive integer that divides exactly two elements of $M$?
b) What is the largest positive integer that divides exactly two elements of $M$?
2006 All-Russian Olympiad Regional Round, 11.7
Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.
1992 Tournament Of Towns, (354) 3
Consider the sequence $a(n)$ defined by the following conditions:$$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$ How many perfect squares no greater in value than $1000 000$ will be found among the first terms of the sequence? ( (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
2008 Thailand Mathematical Olympiad, 8
Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers
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]
2010 Dutch IMO TST, 3
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.
2007 Cuba MO, 2
Find three different positive integers whose sum is minimum than meet the condition that the sum of each pair of them is a perfect square.
1988 Bundeswettbewerb Mathematik, 1
For the natural numbers $x$ and $y$, $2x^2 + x = 3y^2 + y$ .
Prove that then $x-y$, $2x + 2y + 1$ and $3x + 3y + 1$ are perfect squares.
2020 Germany Team Selection Test, 3
Let $a$ and $b$ be two positive integers. Prove that the integer
\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]
is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.)
[i]Russia[/i]
1984 All Soviet Union Mathematical Olympiad, 387
The $x$ and $y$ figures satisfy a condition: for every $n\ge1$ the number $$xx...x6yy...y4$$ ($n$ times $x$ and $n$ times $y$) is a perfect square. Find all possible $x$ and $y$.
1995 Tournament Of Towns, (475) 3
The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square?
(A Tolpygo)
1997 Bundeswettbewerb Mathematik, 4
Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.
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.
1992 Romania Team Selection Test, 2
For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix}
a^2+1 & a \\
a & 1
\end{pmatrix}^{n-2}$
2019 Regional Olympiad of Mexico Center Zone, 1
Let $a$, $b$, and $c $ be integers greater than zero. Show that the numbers $$2a ^ 2 + b ^ 2 + 3 \,\,, 2b ^ 2 + c ^ 2 + 3\,\,, 2c ^ 2 + a ^ 2 + 3 $$ cannot be all perfect squares.