Olympiad problems

2010 Saudi Arabia BMO TST, 1

Find all integers $n$ for which $9n + 16$ and $16n + 9$ are both perfect squares.

2004 Germany Team Selection Test, 3

Tags: perfect square , decimal representation , modular arithmetic , number theory

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

1969 IMO Shortlist, 62

Tags: perfect square , number theory

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

2017 QEDMO 15th, 3

Tags: perfect square , number theory

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 .

2005 Slovenia Team Selection Test, 3

Tags: perfect square , number theory

Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.

1973 All Soviet Union Mathematical Olympiad, 175

Tags: perfect square , 9-digit , number theory

Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.

1977 Spain Mathematical Olympiad, 4

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

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

2005 Argentina National Olympiad, 4

Tags: perfect square , 3d geometry , perfect cube , number theory , geometry

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

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.

2020 Greece JBMO TST, 3

Tags: perfect square , number theory

Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.

2018 Polish Junior MO Finals, 1

Tags: perfect square , power , number theory

Positive odd integers $a, b$ are such that $a^bb^a$ is a perfect square. Show that $ab$ is a perfect square.

1998 Akdeniz University MO, 1

Tags: perfect square , equation , number theory

Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$ is not a perfect square.

1988 IMO Longlists, 14

Tags: divisibility , diophantine equation , perfect square , vieta jumping , number theory

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2016 Singapore Junior Math Olympiad, 1

Tags: perfect square , number theory

Find all integers$ n$ such that $n^2 + 24n + 35$ is a square.

1949-56 Chisinau City MO, 6

Tags: perfect square , remainder , number theory

Prove that the remainder of dividing the square of an integer by $3$ is different from $2$.

1998 Singapore Team Selection Test, 3

Tags: perfect square , perfect cube , modular arithmetic , arithmetic sequence , number theory

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

1996 IMO, 4

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

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

2013 Kyiv Mathematical Festival, 2

Tags: perfect square , sum , representation , number theory

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

2024 JBMO TST - Turkey, 5

Tags: perfect square , divisibility , number theory

Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.

2015 Puerto Rico Team Selection Test, 6

Tags: perfect square , number theory

Find all positive integers $n$ such that $7^n + 147$ is a perfect square.

1997 Abels Math Contest (Norwegian MO), 1

Tags: perfect square , sum , number theory

We call a positive integer $n$ [i]happy [/i] if there exist integers $a,b$ such that $a^2+b^2 = n$. If $t$ is happy, show that (a) $2t$ is [i]happy[/i], (b) $3t$ is not [i]happy[/i]

1979 IMO Shortlist, 21

Tags: perfect square , additive number theory , perfect cube , counting , number theory

Let $N$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3 + 1\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$

2018 May Olympiad, 1

Tags: digit , perfect square , number theory

You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?

2011 Grand Duchy of Lithuania, 1

Tags: perfect square , number theory

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?

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: perfect square , number theory

Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.