Olympiad problems

2014 Saudi Arabia Pre-TST, 1.3

Find all positive integers $n$ for which $1 - 5^n + 5^{2n+1}$ is a perfect square.

2019 China Western Mathematical Olympiad, 1

Tags: number theory , perfect square

Determine all the possible positive integer $n,$ such that $3^n+n^2+2019$ is a perfect square.

2008 Singapore Senior Math Olympiad, 2

Tags: number theory , perfect square

Determine all primes $p$ such that $5^p + 4 p^4$ is a perfect square, i.e., the square of an integer.

2010 Austria Beginners' Competition, 1

Tags: number theory , difference of squares , perfect square

Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

1996 Bundeswettbewerb Mathematik, 4

Tags: number theory , perfect square

Find all natural numbers $n$ for which $n2^{n-1} +1$ is a perfect square.

2020 Greece Team Selection Test, 4

Tags: number theory , perfect square , ceiling function

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

2016 Cono Sur Olympiad, 1

Tags: number theory , perfect square

Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers. [b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.

2004 Switzerland Team Selection Test, 2

Tags: number theory , perfect square

Find the largest natural number $n$ for which $4^{995} +4^{1500} +4^n$ is a square.

2022 AMC 12/AHSME, 16

Tags: perfect square

A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? $\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $

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 $$p^2 + p +1=(r^2 + r + 1)(q^2 + q + 1).$$

2016 Costa Rica - Final Round, A2

Tags: number theory , perfect square

The initial number of inhabitants of a city of more than $150$ inhabitants is a perfect square. With an increase of $1000$ inhabitants it becomes a perfect square plus a unit. After from another increase of $1000$ inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.

2014 Taiwan TST Round 3, 2

Tags: number theory , perfect square , decimal representation

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

2022 Romania Team Selection Test, 4

Tags: number theory , perfect square

Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?

2012 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , perfect square

Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer

1966 All Russian Mathematical Olympiad, 074

Tags: number theory , perfect square

Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?

2007 Estonia Math Open Junior Contests, 10

Tags: number theory , sum , perfect square , combinatorics

Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.

1966 Swedish Mathematical Competition, 3

Tags: number theory , sum of squares , perfect square

Show that an integer $= 7 \mod 8$ cannot be sum of three squares.

2011 May Olympiad, 2

Tags: perfect square , digit , perfect cube , number theory

Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.

1992 Tournament Of Towns, (333) 1

Tags: perfect square , number theory

Prove that the product of all integers from $2^{1917} +1$ up to $2^{1991} -1$ is not the square of an integer. (V. Senderov, Moscow)

2015 Thailand Mathematical Olympiad, 8

Tags: perfect square , odd , number theory

Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.

1974 Swedish Mathematical Competition, 6

Tags: number theory , perfect square , sum of squares

For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that \[ a_1^2+a_2^2+\cdots+a_n^2 \] is a square?

2000 Belarus Team Selection Test, 6.2

Tags: digit , perfect square , number theory

A positive integer $A_k...A_1A_0$ is called monotonic if $A_k \le ..\le A_1 \le A_0$. Show that for any $n \in N$ there is a monotonic perfect square with $n$ digits.

2020 Czech and Slovak Olympiad III A, 4

Tags: number theory , perfect square

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$. Prove that $b$ is a square of a positive integer. (Patrik Bak)

1995 Tournament Of Towns, (475) 3

Tags: perfect square , number theory

The first digit of a $6$-digit number is $5$. Is it true that it is always possible to write $6$ more digits to the right of this number so that the resulting $12$-digit number is a perfect square? (A Tolpygo)

2019 Saudi Arabia Pre-TST + Training Tests, 3.3

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

Define sequence of positive integers $(a_n)$ as $a_1 = a$ and $a_{n+1} = a^2_n + 1$ for $n \ge 1$. Prove that there is no index $n$ for which $$\prod_{k=1}^{n} \left(a^2_k + a_k + 1\right)$$ is a perfect square.