Found problems: 521
2012 Bosnia And Herzegovina - Regional Olympiad, 4
Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer
1995 Mexico National Olympiad, 4
Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square.
Show that one cannot find $27$ such elements.
2011 Vietnam Team Selection Test, 5
Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.
1949-56 Chisinau City MO, 4
Prove that the product of four consecutive integers plus $1$ is a perfect square.
2020 Taiwan TST Round 2, 2
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]
1964 Dutch Mathematical Olympiad, 3
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.
2005 Slovenia Team Selection Test, 3
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
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.
2018 India PRMO, 15
Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?
2017 Balkan MO Shortlist, N2
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
2010 Hanoi Open Mathematics Competitions, 8
If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, fi nd $n$.
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]
1989 Austrian-Polish Competition, 9
Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.
2017 Hanoi Open Mathematics Competitions, 3
Suppose $n^2 + 4n + 25$ is a perfect square. How many such non-negative integers $n$'s are there?
(A): $1$ (B): $2$ (C): $4$ (D): $6$ (E): None of the above.
1997 Tournament Of Towns, (562) 3
All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is:
(a) an integer;
(b) the square of an integer.
(A Kanel)
2014 India IMO Training Camp, 2
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
2002 Belarusian National Olympiad, 1
Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.)
(E. Barabanov, I. Voronovich)
2014 Taiwan TST Round 3, 2
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
1996 Bundeswettbewerb Mathematik, 4
Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.
2019 Portugal MO, 3
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
2018 Azerbaijan BMO TST, 2
Find all functions $f :Z_{>0} \to Z_{>0}$ such that the number $xf(x) + f ^2(y) + 2xf(y)$ is a perfect square for all positive integers $x,y$.
1996 Tournament Of Towns, (484) 2
Does there exist an integer n such that all three numbers
(a) $n - 96$, $n$ and $n + 96$
(b) $n - 1996$, $n$ and $n + 1996$
are positive prime numbers?
(V Senderov)
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.
1992 Tournament Of Towns, (348) 6
Consider the sequence $a(n)$ defined by the following conditions: $$a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...$$
Prove that the sequence contains an infinite number of perfect squares. (Note: $[x]$ means the integer part of $x$, that is the greatest integer not greater than $x$.)
(A Andjans)
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 \( x \)?