Olympiad problems

1917 Eotvos Mathematical Competition, 2

In the square of an integer $ a$, the tens’ digit is $7$. What is the units’ digit of $a^2$?

2021 Austrian MO National Competition, 6

Tags: number theory , Perfect Square , Perfect Squares

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

2019 Durer Math Competition Finals, 3

Tags: Perfect Square , number theory

Determine all triples $(p, q, r)$ of prime numbers for which $p^q + p^r$ is a perfect square.

2016 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , Divisibility , FBH , Perfect Square

Let $a$ and $b$ be two positive integers such that $2ab$ divides $a^2+b^2-a$. Prove that $a$ is perfect square

1989 Spain Mathematical Olympiad, 4

Tags: number theory , power of number , Perfect Square

Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

2018 Dutch IMO TST, 3

Tags: number theory , Sum , floor function , Perfect Square

Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is defined as follows: we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer. Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.

2005 Singapore Senior Math Olympiad, 1

Tags: Perfect Square , Digits , Difference , number theory

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.

1994 Spain Mathematical Olympiad, 1

Tags: Arithmetic Progression , number theory , Perfect Squares , Perfect Square

Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.

1979 IMO Shortlist, 21

Tags: number theory , perfect cube , Perfect Square , counting , Additive Number Theory , IMO Shortlist , IMO Longlist

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 $N >M.$

2013 IMO Shortlist, N4

Tags: number theory , Perfect Square , decimal representation , IMO Shortlist , Hi

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.

2020 Greece Team Selection Test, 4

Tags: ceiling function , number theory , IMO Shortlist , IMO Shortlist 2019 , Perfect Square

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]

1988 Tournament Of Towns, (168) 1

Tags: number theory , Perfect Square

We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number. (Folklore)

1989 Austrian-Polish Competition, 6

Tags: Sequence , prime , Perfect Square , number theory

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

2010 Dutch IMO TST, 3

Tags: number theory , Perfect Square

(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.

2013 Kyiv Mathematical Festival, 2

Tags: number theory , Perfect Square , representation , Sum

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 $2n$?

1979 IMO Longlists, 66

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.

1995 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , Perfect Square

Let $n$ and $p$ be two integers with $p$ positive prime, such that $pn + 1$ is a perfect square. Show that $n + 1$ is the sum of $p$ perfect squares, not necessarily distinct.

2006 Junior Balkan Team Selection Tests - Romania, 4

Tags: Digits , number theory , Perfect Square

For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.

2019 New Zealand MO, 4

Tags: number theory , Perfect Square

Show that for all positive integers $k$, there exists a positive integer n such that $n2^k -7$ is a perfect square.

2012 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , Pythagorean Triple , Perfect Square

Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$. Show that the number $\frac12(c - a) (c - b)$ is square of an integer.

2004 Paraguay Mathematical Olympiad, 5

Tags: number theory , Perfect Square , Digits

We have an integer $A$ such that $A^2$ is a four digit number, with $5$ in the ten's place . Find all possible values of $A$.

1953 Kurschak Competition, 2

Tags: number theory , Perfect Square , divides

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

1949-56 Chisinau City MO, 6

Tags: number theory , Perfect Square , remainder

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

2016 Estonia Team Selection Test, 11

Tags: number theory , Perfect Square

Find all positive integers $n$ such that $(n^2 + 11n - 4) \cdot n! + 33 \cdot 13^n + 4$ is a perfect square

2000 Abels Math Contest (Norwegian MO), 1a

Tags: odd , number theory , Perfect Square

Show that any odd number can be written as the difference between two perfect squares.