Olympiad problems

2023 Indonesia MO, 3

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.

1980 Spain Mathematical Olympiad, 6

Tags: Perfect Square , consecutive , number theory

Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.

2019 Canadian Mathematical Olympiad Qualification, 4

Tags: partition , number theory , Subsets , 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.

1963 All Russian Mathematical Olympiad, 036

Tags: number theory , Perfect Square

Given the endless arithmetic progression with the positive integer members. One of those is an exact square. Prove that the progression contain the infinite number of the exact squares.

2013 India PRMO, 14

Tags: number theory , Perfect Squares , 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 $m + n$?

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?

2012 Balkan MO Shortlist, N1

Tags: number theory , Sequence , Perfect Squares , consecutive , Perfect Square

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.

1996 Tournament Of Towns, (496) 3

Tags: factorial , number theory , Perfect Square

Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square? (S Tokarev)

2009 IMO Shortlist, 7

Tags: number theory , Perfect Square , IMO Shortlist , exponential

Let $a$ and $b$ be distinct integers greater than $1$. Prove that there exists a positive integer $n$ such that $(a^n-1)(b^n-1)$ is not a perfect square. [i]Proposed by Mongolia[/i]

2009 Thailand Mathematical Olympiad, 1

Tags: number theory , Perfect Square

Let $S \subset Z^+$ be a set of positive integers with the following property: for any $a, b \in S$, if $a \ne b$ then $a + b$ is a perfect square. Given that $2009 \in S$ and $2087 \in S$, what is the maximum number of elements in $S$?

1989 Austrian-Polish Competition, 9

Tags: Sum of Squares , Perfect Square , odd , number theory

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

1979 IMO Shortlist, 23

Tags: number theory , Perfect Square , IMO Shortlist , IMO 1979

Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

2017 JBMO Shortlist, NT3

Tags: number theory , Perfect Square , positive integers

Find all pairs of positive integers $(x,y)$ such that $2^x + 3^y$ is a perfect square.

2017 Switzerland - Final Round, 7

Tags: number theory , Perfect Square , diophantine , Diophantine equation

Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

2011 Belarus Team Selection Test, 1

Tags: number theory , Perfect Squares , Perfect Square

Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square. N. Sedrakian , I.Voronovich

1998 Belarus Team Selection Test, 2

Tags: number theory , Perfect Square , diophantine , Diophantine equation

a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

2019 Peru MO (ONEM), 1

Tags: number theory , Perfect Squares , 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.

2004 Federal Competition For Advanced Students, P2, 4

Tags: Perfect Square , Sequence , recurrence relation , number theory

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.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: number theory , divides , prime , Perfect Square

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

1991 All Soviet Union Mathematical Olympiad, 539

Tags: number theory , Perfect Square

Find unequal integers $m, n$ such that $mn + n$ and $mn + m$ are both squares. Can you find such integers between $988$ and $1991$?

2012 Argentina National Olympiad, 4

Tags: number theory , Perfect Square

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: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

2017 CHMMC (Fall), 5

Tags: number theory , Perfect Square , CHMMC , Perfect Squares

Find the number of primes $p$ such that $p! + 25p$ is a perfect square.

2002 Portugal MO, 5

Tags: geometry , Squares , 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]

1995 Mexico National Olympiad, 4

Tags: number theory , Perfect Square

Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square. Show that one cannot find $27$ such elements.

2022 Latvia Baltic Way TST, P13

Tags: number theory , Perfect Square

Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square. Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.