Found problems: 521
1989 Austrian-Polish Competition, 6
A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.
2006 Junior Balkan Team Selection Tests - Romania, 4
For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.
1994 Spain Mathematical Olympiad, 1
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
2021 Middle European Mathematical Olympiad, 8
Prove that there are infinitely many positive integers $n$ such that $n^2$ written in base $4$ contains only digits $1$ and $2$.
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)
2009 Dutch Mathematical Olympiad, 2
Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$.
Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.
V Soros Olympiad 1998 - 99 (Russia), 9.8
Find all natural numbers whose decimal notation consists of different digits of the same parity and which are perfect squares.
2021 South East Mathematical Olympiad, 4
For positive integer $k,$ we say that it is a [i]Taurus integer[/i] if we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square.
$(1)$ Determine whether $2021$ is a Taurus integer.
$(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$
2020 Peru Iberoamerican Team Selection Test, P4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
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
2017 Saudi Arabia JBMO TST, 6
Find all pairs of prime numbers $(p, q)$ such that $p^2 + 5pq + 4q^2$ is a perfect square.
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]
2011 Belarus Team Selection Test, 1
Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square.
N. Sedrakian , I.Voronovich
1996 Tournament Of Towns, (496) 3
Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square?
(S Tokarev)
2021 Polish Junior MO First Round, 1
Is there a six-digit number where every two consecutive digits make up a certain number two-digit number that is the square of an integer? Justify your answer.
1986 IMO, 1
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
2003 Paraguay Mathematical Olympiad, 3
Today the age of Pedro is written and then the age of Luisa, obtaining a number of four digits that is a perfect square. If the same is done in $33$ years from now, there would be a perfect square of four digits . Find the current ages of Pedro and Luisa.
2010 All-Russian Olympiad Regional Round, 10.7
Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?
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.
1994 Greece National Olympiad, 1
Prove that number $2(1991m^2+1993mn+1995n^2)$ where $m,n$ are poitive integers, cannot be a square of an integer.
2014 Belarus Team Selection Test, 3
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.
1997 German National Olympiad, 1
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
2019 Canadian Mathematical Olympiad Qualification, 4
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
1978 Czech and Slovak Olympiad III A, 6
Show that the number
\[p_n=\left(\frac{3+\sqrt5}{2}\right)^n+\left(\frac{3-\sqrt5}{2}\right)^n-2\]
is a positive integer for any positive integer $n.$ Furthermore, show that the numbers $p_{2n-1}$ and $p_{2n}/5$ are perfect squares $($for any positive integer $n).$
2012 Flanders Math Olympiad, 2
Let $n$ be a natural number. Call $a$ the smallest natural number you need to subtract from $n$ to get a perfect square. Call $b$ the smallest natural number that you must add to $n$ to get a perfect square. Prove that $n - ab$ is a perfect square.