Olympiad problems

2021 Durer Math Competition Finals, 2

In a french village the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $ 1$ bigger than a perfect square. If again $100$ more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?

2021 Saudi Arabia Training Tests, 38

Tags: perfect square , number theory

Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair, one number is divided by another.

2011 Grand Duchy of Lithuania, 1

Tags: number theory , perfect square

Integers $a, b$ and $c$ satisfy the condition $ab + bc + ca = 1$. Is it true that the number $(1+a^2)(1+b^2)(1+c^2)$ is a perfect square? Why?

2005 Abels Math Contest (Norwegian MO), 1a

Tags: number theory , sum , perfect square

A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.

1991 IMO Shortlist, 14

Tags: modular arithmetic , number theory , quadratic , perfect square , discriminant

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.$

1999 Switzerland Team Selection Test, 10

Tags: product , consecutive , perfect square , number theory

Prove that the product of five consecutive positive integers cannot be a perfect square.

1931 Eotvos Mathematical Competition, 2

Tags: number theory , odd , perfect square

Let $a^2_1+ a^2_2+ a^2_3+ a^2_4+ a^2_5= b^2$, where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$, and $b$ are integers. Prove that not all of these numbers can be odd.

1984 Bundeswettbewerb Mathematik, 3

Tags: number theory , perfect square

Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.

2019 Canadian Mathematical Olympiad Qualification, 4

Tags: partition , number theory , subset , perfect square

Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, find the largest positive integer $m$ for which such a partition exists.

2010 All-Russian Olympiad Regional Round, 11.4

Tags: number theory , perfect square

We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.

Kvant 2023, M2731

Tags: perfect square , number theory

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]

1986 IMO, 1

Tags: perfect square , quadratic , modular arithmetic , number theory , system of equations

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

OIFMAT III 2013, 1

Tags: number theory , digit , perfect square

Find all four-digit perfect squares such that: $\bullet$ All your figures are less than $9$. $\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.

2007 Thailand Mathematical Olympiad, 4

Tags: number theory , perfect square

Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.

1990 IMO Shortlist, 16

Tags: perfect square , complex number , algebra , convex polygon

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.

2017 Argentina National Olympiad, 4

Tags: perfect square , perfect cube , number theory

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 $1^3,2^3,3^3,4^3,…$ .

2010 Thailand Mathematical Olympiad, 1

Tags: number theory , perfect square

Show that, for every positive integer $x$, there is a positive integer $y\in \{2, 5, 13\}$ such that $xy - 1$ is not a perfect square.

1989 Spain Mathematical Olympiad, 4

Tags: number theory , power of number , perfect square

Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

2020-21 IOQM India, 8

Tags: digit , number theory , perfect square

A $5$-digit number (in base $10$) has digits $k, k + 1, k + 2, 3k, k + 3$ in that order, from left to right. If this number is $m^2$ for some natural number $m$, find the sum of the digits of $m$.

2019 Girls in Mathematics Tournament, 1

Tags: number theory , perfect square , combinatorics

During the factoring class, Esmeralda observed that $1$, $3$ and $5$ can be written as the difference of two perfect squares, as can be seen: $1 = 1^2 - 0^2$ $3 = 2^2 - 1^2$ $5 = 3^2 - 2^2$ a) Show that all numbers written in the form $2 * m + 1$ can be written as a difference of two perfect squares. b) Show how to calculate the value of the expression $E = 1 + 3 + 5 + ... + (2m + 1)$. c) Esmeralda, happy with what she discovered, decided to look for other ways to write $2019$ as the difference of two perfect squares of positive integers. Determine how many ways it can do what you want.

2011 District Olympiad, 2

Tags: number theory , perfect square

a) Show that $m^2- m +1$ is an element of the set $\{n^2 + n +1 | n \in N\}$, for any positive integer $ m$. b) Let $p$ be a perfect square, $p> 1$. Prove that there exists positive integers $r$ and $q$ such that $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

2009 Abels Math Contest (Norwegian MO) Final, 1a

Tags: perfect square , number theory

Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.

2015 Abels Math Contest (Norwegian MO) Final, 4

Tags: number theory , exponential , perfect square

a. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is even. b. Determine all nonnegative integers $x$ and $y$ so that $3^x + 7^y$ is a perfect square and $y$ is odd

2008 Brazil Team Selection Test, 4

Tags: perfect square , number theory , phi function , euler s phi function

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2019 Nigerian Senior MO Round 4, 4

Tags: perfect square , odd , sequence , recurrence relation , algebra , number theory

We consider the real sequence ($x_n$) defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2 x_{n}$ for $n=0,1,2,...$ We define the sequence ($y_n$) by $y_n=x^2_n+2^{n+2}$ for every nonnegative integer $n$. Prove that for every $n>0, y_n$ is the square of an odd integer.