Found problems: 521
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 $\{1,2,\cdots,n\}.$
2011 QEDMO 9th, 1
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares.
1990 IMO Longlists, 58
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
1996 Denmark MO - Mohr Contest, 4
Regarding a natural number $n$, it is stated that the number $n^2$ has $7$ as the second to last digit. What is the last digit of $n^2$?
2023 Indonesia MO, 3
A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero.
Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.
ICMC 6, 4
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
2016 India PRMO, 16
For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.
2021 IMO, 1
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
1992 Tournament Of Towns, (352) 1
Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square.
(S Tokarev)
2012 Hanoi Open Mathematics Competitions, 7
Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)
2021 Czech-Polish-Slovak Junior Match, 4
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.
2024-IMOC, N1
Proof that for every primes $p$, $q$
\[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]
is never a perfect square.
[i]Proposed by chengbilly[/i]
2020 Greece Team Selection Test, 4
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]
2018 IMAR Test, 4
Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square.
[i]* * *[/i]
2010 Austria Beginners' Competition, 1
Prove that $2010$ cannot be represented as the difference between two square numbers.
(B. Schmidt, Graz University of Technology)
1990 IMO, 3
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
2021 Argentina National Olympiad, 5
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.
2020 Peru Iberoamerican Team Selection Test, P4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2012 Argentina National Olympiad, 4
For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$
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.
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.
2016 Singapore Junior Math Olympiad, 1
Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.
2012 Mathcenter Contest + Longlist, 2 sl11
Define the sequence of positive prime numbers. $p_1,p_2,p_3,...$. Let set $A$ be the infinite set of positive integers whose prime divisor does not exceed $p_n$. How many at least members must be selected from the set $A$ , such that we ensures that there are $2$ numbers whose products are perfect squares?
[i](PP-nine)[/i]
1994 All-Russian Olympiad Regional Round, 9.7
Find all prime numbers $p,q,r,s$ such that their sum is a prime number and $p^2+qs$ and $p^2 +qr$ are squares of integers.
1974 Swedish Mathematical Competition, 6
For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that
\[
a_1^2+a_2^2+\cdots+a_n^2
\]
is a square?