Found problems: 171
2021 Abels Math Contest (Norwegian MO) Final, 3a
For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ 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.
2018 Bosnia And Herzegovina - Regional Olympiad, 2
Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer
2017 South East Mathematical Olympiad, 7
Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$
2023 239 Open Mathematical Olympiad, 4
We call a natural number [i]almost a square[/i] if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.
2019 Tournament Of Towns, 5
Let us say that the pair $(m, n)$ of two positive different integers m and n is [i]nice [/i] if $mn$ and $(m + 1)(n + 1)$ are perfect squares. Prove that for each positive integer m there exists at least one $n > m$ such that the pair $(m, n)$ is nice.
(Yury Markelov)
2018 Malaysia National Olympiad, A5
Determine the value of
$(101 \times 99)$ - $(102 \times 98)$ + $(103 \times 97)$ − $(104 \times 96)$ + ... ... + $(149 \times 51)$ − $(150 \times 50)$.
1992 IMO, 3
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 $S(n)=n^{2}-14.$
2020 Olympic Revenge, 4
Let $n$ be a positive integer and $A$ a set of integers such that the set $\{x = a + b\ |\ a, b \in A\}$ contains $\{1^2, 2^2, \dots, n^2\}$. Prove that there is a positive integer $N$ such that if $n \ge N$, then $|A| > n^{0.666}$.
1998 Bundeswettbewerb Mathematik, 2
Prove that there exists an infinite sequence of perfect squares with the following properties:
(i) The arithmetic mean of any two consecutive terms is a perfect square,
(ii) Every two consecutive terms are coprime,
(iii) The sequence is strictly increasing.
2020 Kosovo National Mathematical Olympiad, 2
Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.
2017 Ecuador NMO (OMEC), 5
Let the sequences $(x_n)$ and $(y_n)$ be defined by $x_0 = 0$, $x_1 = 1$, $x_{n + 2} = 3x_{n + 1}-2x_n$ for $n = 0, 1, ...$ and $y_n = x^2_n+2^{n + 2}$ for $n = 0, 1, ...,$ respectively. Show that for all n> 0, and n is the square of a odd integer.
2014 Finnish National High School Mathematics, 5
Determine the smallest number $n \in Z_+$, which can be written as $n = \Sigma_{a\in A}a^2$, where $A$ is a finite set of positive integers and $\Sigma_{a\in A}a= 2014$.
In other words: what is the smallest positive number which can be written as a sum of squares of different positive integers summing to $2014$?
OIFMAT III 2013, 4
Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?
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.
2021 Malaysia IMONST 1, 15
Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.
2021 Argentina National Olympiad, 2
Let $m$ be a positive integer for which there exists a positive integer $n$ such that the multiplication $mn$ is a perfect square and $m- n$ is prime. Find all $m$ for $1000\leq m \leq 2021.$
2015 Taiwan TST Round 2, 1
Let the sequence $\{a_n\}$ satisfy $a_{n+1}=a_n^3+103,n=1,2,...$. Prove that at most one integer $n$ such that $a_n$ is a perfect square.
1986 IMO Longlists, 26
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.
2016 Poland - Second Round, 4
Let $k$ be a positive integer. Show that exists positive integer $n$, such that sets $A = \{ 1^2, 2^2, 3^2, ...\}$ and $B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}$ have exactly $k$ common elements.
2018 May Olympiad, 1
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?
1992 IMO Longlists, 22
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 $S(n)=n^{2}-14.$
2003 IMO Shortlist, 5
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]
2020 Federal Competition For Advanced Students, P1, 4
Determine all positive integers $N$ such that
$$2^N-2N$$
is a perfect square.
(Walther Janous)
2020 Regional Olympiad of Mexico West, 3
Prove that for every natural number \( n>2 \) there exists an integer \( k \) that can be written as the sum of \( i \) positive perfect squares, for every \( i \) between \( 2 \) and \( n \).