Olympiad problems

1997 Bundeswettbewerb Mathematik, 4

Prove that if $n$ is a natural number such that both $3n+1$ and $4n+1$ are squares, then $n$ is divisible by $56$.

2022 Romania Team Selection Test, 4

Tags: number theory , Perfect Squares , romania , Romanian TST

Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?

2020 Dutch Mathematical Olympiad, 5

Tags: combinatorics , number theory , Perfect Squares

Sabine has a very large collection of shells. She decides to give part of her collection to her sister. On the first day, she lines up all her shells. She takes the shells that are in a position that is a perfect square (the first, fourth, ninth, sixteenth, etc. shell), and gives them to her sister. On the second day, she lines up her remaining shells. Again, she takes the shells that are in a position that is a perfect square, and gives them to her sister. She repeats this process every day. The $27$th day is the first day that she ends up with fewer than $1000$ shells. The $28$th day she ends up with a number of shells that is a perfect square for the tenth time. What are the possible numbers of shells that Sabine could have had in the very beginning?

2023 Stars of Mathematics, 1

Tags: number theory , prime numbers , Perfect Squares

Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

1969 IMO Longlists, 62

Tags: number theory , Perfect Squares , IMO Shortlist , IMO Longlist

Which natural numbers can be expressed as the difference of squares of two integers?

2011 Vietnam Team Selection Test, 5

Tags: induction , modular arithmetic , number theory , Perfect Squares

Find all positive integers $n$ such that $A=2^{n+2}(2^n-1)-8\cdot 3^n +1$ is a perfect square.

1990 IMO Longlists, 58

Tags: complex numbers , algebra , convex polygon , Perfect Squares , IMO , IMO 1990 , Harm Derksen

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.

2020 Malaysia IMONST 1, 14

Tags: Perfect Squares

A perfect square ends with the same two digits. How many possible values of this digit are there?

2022 Korea National Olympiad, 3

Tags: number theory , Perfect Squares , Sequence , Integer sequence

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.

2021 South Africa National Olympiad, 3

Tags: number theory , Perfect Squares , Sum of Squares , primes , power of 2

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

2013 Dutch IMO TST, 2

Tags: Perfect Squares , number theory , rational

Determine all integers $n$ for which $\frac{4n-2}{n+5}$ is the square of a rational number.

Russian TST 2015, P3

Tags: number theory , Perfect Squares

Find all integers $k{}$ for which there are infinitely many triples of integers $(a,b,c)$ such that \[(a^2-k)(b^2-k)=c^2-k.\]

2002 Brazil National Olympiad, 1

Tags: number theory , Perfect Squares , perfect cube , Perfect Powers , Subset , Sum

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

2025 All-Russian Olympiad, 9.3

Tags: number theory , Perfect Squares

Find all natural numbers $n$ for which there exists an even natural number $a$ such that the number \[ (a - 1)(a^2 - 1)\cdots(a^n - 1) \] is a perfect square.

2010 USAJMO, 1

Tags: AMC , USAJMO , Perfect Squares , permutations , USA(J)MO

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

2010 Contests, 3

Tags: function , algebra , IMO Shortlist , IMO 2010 , Perfect Squares , number theory , IMO

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]

2022 Saudi Arabia JBMO TST, 1

Tags: number theory , Perfect Squares

The positive $n > 3$ called ‘nice’ if and only if $n +1$ and $8n + 1$ are both perfect squares. How many positive integers $k \le 15$ such that $4n + k$ are composites for all nice numbers $n$?

2010 Hanoi Open Mathematics Competitions, 8

Tags: number theory , Perfect Squares , Perfect Square

If $n$ and $n^3+2n^2+2n+4$ are both perfect squares, find $n$.

2001 China Team Selection Test, 1

Tags: number theory , Perfect Squares

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 $?

2011 District Olympiad, 3

Tags: number theory , Perfect Squares , Perfect Square

A positive integer $N$ has the digits $1, 2, 3, 4, 5, 6$ and $7$, so that each digit $i$, $i \in \{1, 2, 3, 4, 5, 6, 7\}$ occurs $4i$ times in the decimal representation of $N$. Prove that $N$ is not a perfect square.

2020 Romania EGMO TST, P3

Tags: algebra , Perfect Squares , number theory , romania

The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares. [i]George Stoica, Canada[/i]

1987 Greece National Olympiad, 2

Tags: number theory , Perfect Squares , Perfect Square

Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .

2010 Contests, 1

Tags: AMC , USAJMO , Perfect Squares , permutations , USA(J)MO

A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.

2010 Austria Beginners' Competition, 1

Tags: number theory , Perfect Squares , difference of squares

Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

2018 India PRMO, 15

Tags: minimum , number theory , Perfect Squares , Perfect Square

Let $a$ and $b$ be natural numbers such that $2a-b$, $a-2b$ and $a+b$ are all distinct squares. What is the smallest possible value of $b$ ?