Olympiad problems

1987 Tournament Of Towns, (141) 1

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2009 Hanoi Open Mathematics Competitions, 3

Tags: number theory , perfect square

Let $a, b,c$ be positive integers with no common factor and satisfy the conditions $\frac1a +\frac1b=\frac1c$ Prove that $a + b$ is a square.

2024-IMOC, N1

Tags: number theory , prime , perfect square

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]

2010 Dutch IMO TST, 3

Tags: perfect square , number theory

Let $n\ge 2$ be a positive integer and $p $ a prime such that $n|p-1$ and $p | n^3-1$. Show $ 4p-3$ is a square.

2006 Chile National Olympiad, 4

Tags: number theory , perfect square , perfect cube

Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .

2015 Thailand TSTST, 1

Tags: number theory , square , perfect square , sum of squares

Prove that there exist infinitely many integers $n$ such that $n, n + 1, n + 2$ are each the sum of two squares of integers.

2010 Dutch BxMO TST, 5

Tags: number theory , permutation , perfect square , quadratic

For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.

1940 Moscow Mathematical Olympiad, 062-

Tags: number theory , perfect square

Find a four-digit number that is perfect square and such that the first two digits are the same and the last two as well.

2009 Thailand Mathematical Olympiad, 1

Tags: number theory , perfect square

Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

2020 Kosovo National Mathematical Olympiad, 2

Tags: number theory , perfect square

Find all positive integers $x$, $y$ such that $2^x+5^y+2$ is a perfect square.

2019 Portugal MO, 3

Tags: factorial , perfect square , number theory

The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?

2021 Brazil National Olympiad, 3

Tags: number theory , perfect square , floor function , irrational number , power

Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is a perfect square minus $k$ for every integer $n$ with $n>N$.

2020 Malaysia IMONST 1, 14

Tags: perfect square

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

2023 AMC 10, 15

Tags: perfect square , number theory

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16!$ is a perfect square? $\textbf{(A) }30\qquad\textbf{(B) }30030\qquad\textbf{(C) }70\qquad\textbf{(D) }1430\qquad\textbf{(E) }1001$

2022 Cyprus JBMO TST, 1

Tags: perfect square , number theory

Find all integer values of $x$ for which the value of the expression \[x^2+6x+33\] is a perfect square.

2018 Greece National Olympiad, 1

Tags: sequence , perfect square

Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$ and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.

2013 NZMOC Camp Selection Problems, 8

Tags: number theory , perfect square

Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

Kvant 2023, M2731

Tags: number theory , perfect square

There are 2023 natural written in a row. The first number is 12, and each number starting from the third is equal to the product of the previous two numbers, or to the previous number increased by 4. What is the largest number of perfect squares that can be among the 2023 numbers? [i]Based on the British Mathematical Olympiad[/i]

2022 China Team Selection Test, 5

Tags: number theory , perfect square

Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.

2011 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , perfect square

For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square

2015 Taiwan TST Round 2, 1

Tags: number theory , perfect square , integer sequence

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.

1938 Eotvos Mathematical Competition, 1

Tags: number theory , perfect square

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

2019 IMO Shortlist, N8

Tags: perfect square , ceiling function , number theory

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]

1979 IMO Longlists, 66

Tags: number theory , perfect square

Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

2017 Argentina National Olympiad, 4

Tags: number theory , perfect square , perfect cube

For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$ Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .