Olympiad problems

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.

2013 India PRMO, 14

Tags: number theory , perfect square

Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?

2020 Romania EGMO TST, P3

Tags: perfect square , algebra , number theory

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]

1994 Greece National Olympiad, 1

Tags: number theory , perfect square

Prove that number $2(1991m^2+1993mn+1995n^2)$ where $m,n$ are poitive integers, cannot be a square of an integer.

2017 QEDMO 15th, 3

Tags: number theory , perfect square

Let $a,b,c$ natural numbers for which $a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2$. Show that $ab, bc, ca$ and $ab + bc + ca$ are perfect squares .

2021 Abels Math Contest (Norwegian MO) Final, 3a

Tags: number theory , perfect square

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?

2000 Moldova Team Selection Test, 9

Tags: perfect square , integer , sequence , number theory with sequences

The sequence $x_{n}$ is defined by: $x_{0}=1, x_{1}=0, x_{2}=1,x_{3}=1, x_{n+3}=\frac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\frac{n+1}{n}x_{n} (n=1,2,3..)$ Prove that all members of the sequence are perfect squares.

2013 Kyiv Mathematical Festival, 2

Tags: number theory , representation , sum , perfect square

For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of n distinct positive integers not exceeding $\frac{3n}{2}$ ?

1996 IMO Shortlist, 3

Tags: perfect square , number theory , integer sequence , extremal combinatorics , recurrence relation

A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$ a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$ b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$

1966 All Russian Mathematical Olympiad, 074

Tags: number theory , perfect square

Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?

2019 Peru MO (ONEM), 1

Tags: number theory , perfect square

Determine for what $n\ge 3$ integer numbers, it is possible to find positive integer numbers $a_1 < a_2 < ...< a_n$ such $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$ and $a_1 a_2\cdot\cdot\cdot a_n$ is a perfect square.

2022 Korea National Olympiad, 3

Tags: perfect square , number theory , 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 $\sum_{k=N}^{N+2021}a_k$.

2002 Portugal MO, 5

Tags: square , geometry , perfect square

Consider the three squares indicated in the figure. Show that if the lengths of the sides of the smaller square and the square greater are integers, then adding to the area of the smallest square the area of the inclined square, a perfect square is obtained. [img]https://1.bp.blogspot.com/-B0QdvZIjOLw/X4URvs3C0ZI/AAAAAAAAMmw/S5zMpPBXBn8Jj39d-OZVtMRUDn3tXbyWgCLcBGAsYHQ/s0/2002%2Bportugal%2Bp5.png[/img]

1999 Switzerland Team Selection Test, 6

Tags: divisible , perfect square , number theory

Prove that if $m$ and $n$ are positive integers such that $m^2 + n^2 - m$ is divisible by $2mn$, then $m$ is a perfect square.

2020-21 IOQM India, 6

Tags: number theory , perfect square

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?

2006 Spain Mathematical Olympiad, 2

Tags: number theory , product , consecutive , perfect square , perfect cube

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

1996 Dutch Mathematical Olympiad, 5

Tags: number theory , greatest common divisor , perfect square

For the positive integers $x , y$ and $z$ apply $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}$ . Prove that if the three numbers $x , y,$ and $z$ have no common divisor greater than $1$, $x + y$ is the square of an integer.

1969 IMO Longlists, 62

Tags: perfect square , number theory

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

2000 IMO Shortlist, 6

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

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

2022 Canadian Mathematical Olympiad Qualification, 2

Tags: number theory , perfect square

Determine all pairs of integers $(m, n)$ such that $m^2 + n$ and $n^2 + m$ are both perfect squares.

2012 Bundeswettbewerb Mathematik, 2

Tags: number theory , perfect square

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

2011 Korea Junior Math Olympiad, 3

Tags: number theory , coprime , perfect square

Let $x, y$ be positive integers such that $gcd(x, y) = 1$ and $x + 3y^2$ is a perfect square. Prove that $x^2 + 9y^4$ can't be a perfect square.

2006 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , sum of squares , perfect square

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , divisibility , fbh , perfect square

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

2021 Durer Math Competition Finals, 2

Tags: number theory , perfect square

Find the number of integers $n$ between $1$ and $2021$ such that $2^n+2^{n+3}$ is a perfect square.