Found problems: 521
2001 China Team Selection Test, 1
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
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 $S(n)=n^{2}-14.$
2019 New Zealand MO, 4
Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.
2021 Malaysia IMONST 1, 5
How many integers $n$ (with $1 \le n \le 2021$) have the property that $8n + 1$ is a perfect square?
Russian TST 2020, P3
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]
2012 Czech-Polish-Slovak Junior Match, 5
Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$.
Show that the number $\frac12(c - a) (c - b)$ is square of an integer.
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.
2007 Dutch Mathematical Olympiad, 4
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.)
2007 Greece JBMO TST, 2
Let $n$ be a positive integer such that $n(n+3)$ is a perfect square of an integer, prove that $n$ is not a multiple of $3$.
2021 Azerbaijan Senior NMO, 2
Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.
1976 Czech and Slovak Olympiad III A, 1
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]
2020 Iran MO (2nd Round), P5
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.
Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.
2024 OMpD, 4
Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties:
- \(a_0\) is a given positive integer;
- For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square.
For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on.
(a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers.
Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite.
(b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).
2009 Bosnia And Herzegovina - Regional Olympiad, 1
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers
2024 Mexican University Math Olympiad, 1
Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.
2019 Regional Olympiad of Mexico West, 3
Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$ is the square of a prime number.
2012 Estonia Team Selection Test, 1
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
2009 IMAR Test, 4
Given any $n$ positive integers, and a sequence of $2^n$ integers (with terms among them), prove there exists a subsequence made of consecutive terms, such that the product of its terms is a perfect square. Also show that we cannot replace $2^n$ with any lower value (therefore $2^n$ is the threshold value for this property).
1991 IMO Shortlist, 14
Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$
2020 Federal Competition For Advanced Students, P1, 4
Determine all positive integers $N$ such that
$$2^N-2N$$
is a perfect square.
(Walther Janous)
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P5
We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares?
[i]Proposed by Nikola Velov[/i]
1992 All Soviet Union Mathematical Olympiad, 570
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
2009 Belarus Team Selection Test, 1
Prove that there exist many natural numbers n so that both roots of the quadratic equation $x^2+(2-3n^2)x+(n^2-1)^2=0$ are perfect squares.
S. Kuzmich
2004 India IMO Training Camp, 3
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]
2021 Olympic Revenge, 5
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.