Olympiad problems

2018 Istmo Centroamericano MO, 4

Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.

1996 IMO Shortlist, 3

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

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.$

1949-56 Chisinau City MO, 5

Tags: perfect square , last digit , number theory

Prove that the square of any integer cannot end with two fives.

2025 Israel National Olympiad (Gillis), P3

Tags: digit , perfect square , number theory

Bart wrote the digit "$1$" $2024$ times in a row. Then, Lisa wrote an additional $2024$ digits to the right of the digits Bart wrote, such that the resulting number is a square of an integer. Find all possibilities for the digits Lisa wrote.

2020 Peru Cono Sur TST., P4

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

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

2012 Argentina National Olympiad, 4

Tags: perfect square , number theory

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

2012 Balkan MO Shortlist, N1

Tags: sequence , consecutive , perfect square , number theory

A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

2020 Junior Balkan Team Selection Tests - Moldova, 10

Tags: prime number , perfect square , number theory

Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.

2023 Indonesia MO, 3

Tags: combinatorics , game theory , perfect square

A natural number $n$ is written on a board. On every step, Neneng and Asep changes the number on the board with the following rule: Suppose the number on the board is $X$. Initially, Neneng chooses the sign up or down. Then, Asep will pick a positive divisor $d$ of $X$, and replace $X$ with $X+d$ if Neneng chose the sign "up" or $X-d$ if Neneng chose "down". This procedure is then repeated. Asep wins if the number on the board is a nonzero perfect square, and loses if at any point he writes zero. Prove that if $n \geq 14$, Asep can win in at most $(n-5)/4$ steps.

2017 Czech-Polish-Slovak Match, 1

Tags: perfect square , positive integer , number theory

Find all positive real numbers $c$ such that there are infinitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer. (Slovakia)

2020-21 IOQM India, 6

Tags: perfect square , number theory

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?

2003 Croatia Team Selection Test, 1

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.

2010 Hanoi Open Mathematics Competitions, 8

Tags: perfect square , number theory

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

2003 Junior Tuymaada Olympiad, 2

Tags: number theory , prime , perfect square

Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.

2010 Dutch IMO TST, 3

Tags: perfect square , number theory

(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer. Prove that $M(a,b)$ is a square. (b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.

2004 Germany Team Selection Test, 3

Tags: decimal representation , perfect square , 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]

2020 Malaysia IMONST 1, 4

Tags: perfect square , number theory

This sequence lists the perfect squares in increasing order: \[0,1,4,9,16,\cdots ,a,10^8,b,\cdots\] Determine the value of $b-a$.

2022 Grand Duchy of Lithuania, 4

Tags: perfect square , number theory

Find all triples of natural numbers $(a, b, c)$ for which the number $$2^a + 2^b + 2^c + 3$$ is the square of an integer.

1989 Brazil National Olympiad, 2

Tags: number theory , pell equation , perfect square

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

1994 Nordic, 4

Tags: pell equation , perfect square , integer , number theory

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

1986 IMO Shortlist, 5

Tags: modular arithmetic , quadratic , perfect square , 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.

2021 Malaysia IMONST 2, 2

Tags: number theory , perfect square

Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?

1966 Dutch Mathematical Olympiad, 3

Tags: perfect square , number theory

How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

2025 All-Russian Olympiad, 9.3

Tags: perfect square , number theory

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.

2011 Saudi Arabia Pre-TST, 2.3

Tags: polynomial , perfect square , algebra

Let $f = aX^2 + bX+ c \in Z[X]$ be a polynomial such that for every positive integer $n$,$ f(n )$ is a perfect square. Prove that $f = g^2$ for some polynomial $g \in Z[X]$.