Olympiad problems

1979 IMO Longlists, 69

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 $N >M.$

2019 New Zealand MO, 5

Tags: number theory , perfect square

Find all positive integers $n$ such that $n^4 - n^3 + 3n^2 + 5$ is a perfect square.

2021 IMO, 1

Tags: algebra , perfect square , number theory , combinatorics

Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.

2004 Estonia National Olympiad, 3

Tags: number theory , digit , perfect square

The teacher had written on the board a positive integer consisting of a number of $4$s followed by the same number of $8$s followed . During the break, Juku stepped up to the board and added to the number one more $4$ at the start and a $9$ at the end. Prove that the resulting number is an a square. of an integer.

2019 Girls in Mathematics Tournament, 1

Tags: combinatorics , perfect square , number theory

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.

2020 Regional Olympiad of Mexico West, 3

Tags: number theory , perfect square

Prove that for every natural number $ n>2 $ there exists an integer $ k $ that can be written as the sum of $ i $ positive perfect squares, for every $ i $ between $ 2 $ and $ n $.

2023 Peru MO (ONEM), 1

Tags: divide , perfect square , number theory

We define the set $M = \{1^2,2^2,3^2,..., 99^2, 100^2\}$. a) What is the smallest positive integer that divides exactly two elements of $M$? b) What is the largest positive integer that divides exactly two elements of $M$?

2022 Cyprus TST, 2

Tags: perfect square , pell equation , number theory

Let $n, m$ be positive integers such that \[n(4n+1)=m(5m+1)\] (a) Show that the difference $n-m$ is a perfect square of a positive integer. (b) Find a pair of positive integers $(n, m)$ which satisfies the above relation. Additional part (not asked in the TST): Find all such pairs $(n,m)$.

2005 iTest, 3

Tags: number theory , probability , perfect square , pascal triangle

Find the probability that any given row in Pascal’s Triangle contains a perfect square. [i] (.1 point)[/i]

1974 Dutch Mathematical Olympiad, 4

Tags: number theory , perfect square

For which $n$ is $n^4+6n^3+11n^2+3n+31$ a perfect square?

1981 Swedish Mathematical Competition, 1

Tags: number theory , perfect square

Let $N = 11\cdots 122 \cdots 25$, where there are $n$ $1$s and $n+1$ $2$s. Show that $N$ is a perfect square.

1992 Romania Team Selection Test, 2

Tags: matrix , linear algebra , perfect square , sequence

For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix} a^2+1 & a \\ a & 1 \end{pmatrix}^{n-2}$

2005 Singapore Senior Math Olympiad, 1

Tags: number theory , perfect square , digit , difference

The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.

2019 Saudi Arabia Pre-TST + Training Tests, 2.1

Tags: number theory , perfect square

Suppose that $a, b, c,d$ are pairwise distinct positive integers such that $a+b = c+d = p$ for some odd prime $p > 3$ . Prove that $abcd$ is not a perfect square.

2004 Federal Competition For Advanced Students, P2, 4

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

Show that there is an infinite sequence $a_1,a_2,...$ of natural numbers such that $a^2_1+a^2_2+ ...+a^2_N$ is a perfect square for all $N$. Give a recurrent formula for one such sequence.

2023 New Zealand MO, 1

Tags: number theory , perfect square

For any positive integer $n$ let $n! = 1\times 2\times 3\times ... \times n$. Do there exist infinitely many triples $(p, q, r)$, of positive integers with $p > q > r > 1$ such that the product $p! \cdot q! \cdot r!$$ is a perfect square?

1945 Moscow Mathematical Olympiad, 093

Tags: number theory , 2-digit , perfect square

Find all two-digit numbers $\overline {ab}$ such that $\overline {ab} + \overline {ba}$ is a perfect square.

2023 Stars of Mathematics, 2

Tags: number theory , perfect square

Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$. [i]Proposed by Vlad Matei[/i]

2023 SG Originals, Q4

Tags: sum of divisors , perfect square , number theory

Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square? [i]Proposed by Dylan Toh[/i]

2005 Austrian-Polish Competition, 4

Tags: number theory , perfect square , fermat's little theorem , prime , diophantine equation

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

1949-56 Chisinau City MO, 4

Tags: consecutive , perfect square , algebra

Prove that the product of four consecutive integers plus $1$ is a perfect square.

2001 Cuba MO, 3

Tags: number theory , perfect square

Let $n$ be a positive integer. a) Prove that the number $(2n + 1)^3 - (2n - 1)^3$ is the sum of three perfect squares. b) Prove that the number $(2n+1)^3-2$ is the sum of $3n-1$ perfect squares greater than $1$.

2018 Polish Junior MO Second Round, 5

Tags: combinatorics , number theory , coloring , perfect square

Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.

2002 Brazil National Olympiad, 1

Tags: number theory , subset , perfect cube , perfect power , sum , perfect square

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

2022 Korea National Olympiad, 3

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