Found problems: 521
1998 Romania National Olympiad, 2
Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.
2018 Istmo Centroamericano MO, 4
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.
2011 Denmark MO - Mohr Contest, 5
Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$.
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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.
2013 Vietnam Team Selection Test, 2
a. Prove that there are infinitely many positive integers $t$ such that both $2012t+1$ and $2013t+1$ are perfect squares.
b. Suppose that $m,n$ are positive integers such that both $mn+1$ and $mn+n+1$ are perfect squares. Prove that $8(2m+1)$ divides $n$.
1999 Bundeswettbewerb Mathematik, 4
A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube.
Prove that if $r$ and $s$ are any two positive integers, then
(a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i],
(b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].
2001 Denmark MO - Mohr Contest, 4
Show that any number of the form
$$4444 ...44 88...8$$
where there are twice as many $4$s as $8$s is a square number.
1977 All Soviet Union Mathematical Olympiad, 244
Let us call "fine" the $2n$-digit number if it is exact square itself and the two numbers represented by its first $n$ digits (first digit may not be zero) and last $n$ digits (first digit may be zero, but it may not be zero itself) are exact squares also.
a) Find all two- and four-digit fine numbers.
b) Is there any six-digit fine number?
c) Prove that there exists $20$-digit fine number.
d) Prove that there exist at least ten $100$-digit fine numbers.
e) Prove that there exists $30$-digit fine number.
2015 Czech-Polish-Slovak Junior Match, 2
Decide if the vertices of a regular $30$-gon can be numbered by numbers $1, 2,.., 30$ in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.
2010 IMO Shortlist, 5
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$
[i]Proposed by Gabriel Carroll, USA[/i]
1992 Chile National Olympiad, 1
Determine all naturals $n$ such that $2^n + 5$ is a perfect square.
1997 All-Russian Olympiad Regional Round, 8.1
Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.
2021 Argentina National Olympiad Level 2, 5
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.
1949-56 Chisinau City MO, 6
Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.
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.
2005 iTest, 34
If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.
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 $k > \ell\ge 0$.
1941 Moscow Mathematical Olympiad, 075
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
2005 Bosnia and Herzegovina Junior BMO TST, 2
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an integer.”
2009 Puerto Rico Team Selection Test, 2
The last three digits of $ N$ are $ x25$. For how many values of $ x$ can $ N$ be the square of an integer?
2005 iTest, 3
Find the probability that any given row in Pascal’s Triangle contains a perfect square.
[i] (.1 point)[/i]
2018 Rio de Janeiro Mathematical Olympiad, 3
Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$.
Prove that the number of good functions is a perfect square.
1998 Belarus Team Selection Test, 2
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
2003 Belarusian National Olympiad, 2
Let $P(x) =(x+1)^p (x-3)^q=x^n+a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$ where $p$ and $q$ are positive integers
a) Given $a_1=a_2$, prove that $3n$ is a perfect square.
b) Prove that there exist infinitely many pairs $(p, q)$ of positive integers p and q such that the equality $a_1=a_2$ is valid for the polynomial $P(x)$.
(D. Bazylev)
1992 IMO Shortlist, 21
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 $S(n)=n^{2}-14.$