Found problems: 397
2016 Estonia Team Selection Test, 11
Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square
2018 BAMO, C/1
An integer $c$ is [i]square-friendly[/i] if it has the following property:
For every integer $m$, the number $m^2+18m+c$ is a perfect square.
(A perfect square is a number of the form $n^2$, where $n$ is an integer. For example, $49 = 7^2$ is a perfect square while $46$ is not a perfect square. Further, as an example, $6$ is not [i]square-friendly[/i] because for $m = 2$, we have $(2)2 +(18)(2)+6 = 46$, and $46$ is not a perfect square.)
In fact, exactly one square-friendly integer exists. Show that this is the case by doing the following:
(a) Find a square-friendly integer, and prove that it is square-friendly.
(b) Prove that there cannot be two different square-friendly integers.
1955 Moscow Mathematical Olympiad, 307
* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.
2009 Abels Math Contest (Norwegian MO) Final, 1a
Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.
2019 Thailand TSTST, 1
Find all primes $p$ such that $(p-3)^p+p^2$ is a perfect square.
India EGMO 2021 TST, 6
Let $n>2$ be a positive integer and $b=2^{2^n}$. Let $a$ be an odd positive integer such that $a\le b \le 2a$.
Show that $a^2+b^2-ab$ is not a square.
2020 Greece JBMO TST, 3
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.
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)
2010 Saudi Arabia Pre-TST, 1.2
Find all integers $n$ for which $n(n + 2010)$ is a perfect square.
2019 Junior Balkan Team Selection Tests - Romania, 1
Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.
1976 Swedish Mathematical Competition, 3
If $a$, $b$, $c$ are rational, show that
\[
\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2}
\]
is the square of a rational.
2005 iTest, 3
Find the probability that any given row in Pascal’s Triangle contains a perfect square.
[i] (.1 point)[/i]
2019 Nigerian Senior MO Round 4, 4
We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$
We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$.
Prove that for every $n>0, y_n$ is the square of an odd integer.
2022 Canadian Mathematical Olympiad Qualification, 2
Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.
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?
1979 IMO Longlists, 69
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
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 $b^2$.
.
2001 Singapore Team Selection Test, 1
Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.
2020 Czech and Slovak Olympiad III A, 4
Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$.
Prove that $b$ is a square of a positive integer.
(Patrik Bak)
1993 Bundeswettbewerb Mathematik, 3
There are pairs of square numbers with the following two properties:
(1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ .
(2) If one appends the second to the decimal representation of the first, the decimal representation results another square number.
Example: $16$ and $81$; $1681 = 41^2$.
Prove that there are infinitely many pairs of squares with these properties.
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2003 Junior Tuymaada Olympiad, 2
Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.
2005 Argentina National Olympiad, 4
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ .
Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners.
Prove that Germán can always achieve his goal.
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.
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.