Olympiad problems

2022 Abelkonkurransen Finale, 1a

Determine all positive integers $n$ such that $2022 + 3^n$ is a perfect square.

2016 India PRMO, 16

Tags: algebra , geometric mean , arithmetic mean , perfect square , number theory

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

1952 Moscow Mathematical Olympiad, 224

Tags: number theory , perfect square

a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,. b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places

1998 Romania National Olympiad, 2

Tags: perfect square , number theory

Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.

2019 Durer Math Competition Finals, 2

Tags: number theory , perfect square

Anne multiplies each two-digit number by $588$ in turn, and writes down the so-obtained products. How many perfect squares does she write down?

2017 Thailand Mathematical Olympiad, 5

Tags: perfect square , sum of squares , consecutive , number theory

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.

2018 Dutch IMO TST, 3

Tags: floor function , sum , perfect square , number theory

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 $k > \ell\ge 0$.

1978 Chisinau City MO, 155

Tags: number theory , perfect square

Find the base of the number system less than $100$, in which $2101$ is a perfect square.

1993 Tournament Of Towns, (360) 3

Tags: diophantine , perfect square , diophantine equation , number theory

Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and $$\frac{ab}{a-b}=c$$ Prove that $a -b$ is a perfect square. (SL Berlov)

1955 Moscow Mathematical Olympiad, 307

Tags: trinomial , perfect square , algebra , number theory

* The quadratic expression $ax^2 + bx + c$ is a square (of an integer) for any integer $x$. Prove that $ax^2 + bx + c = (dx + e)^2$ for some integers d and e.

2012 Bundeswettbewerb Mathematik, 2

Tags: number theory , perfect square

Are there positive integers $a$ and $b$ such that both $a^2 + 4b$ and $b^2 + 4a$ are perfect squares?

1986 IMO Longlists, 26

Tags: number theory , quadratic , perfect square , system of equations , modular arithmetic

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.

2013 Hanoi Open Mathematics Competitions, 2

Tags: perfect square , number theory

How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

2015 Indonesia MO Shortlist, N1

Tags: prime , perfect square , triplets , number theory

A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies: (i) $a> b> c$ are prime numbers (ii) $a = b + 2c$ (iii) $a + b + c$ is a perfect square number Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].

1988 Austrian-Polish Competition, 5

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

Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.

1991 ITAMO, 2

Tags: number theory , perfect square

Prove that no number of the form $a^3+3a^2+a$, for a positive integer $a$, is a perfect square.

2018 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: perfect square , integer , number theory

Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer

2005 Abels Math Contest (Norwegian MO), 1a

Tags: sum , perfect square , number theory

A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.

1997 Tuymaada Olympiad, 1

Tags: number theory , product , perfect square

The product of any three of these four natural numbers is a perfect square. Prove that these numbers themselves are perfect squares.

2015 Czech-Polish-Slovak Junior Match, 2

Tags: number theory , sum , perfect square

Decide if the vertices of a regular $30$-gon can be numbered by numbers $1, 2,.., 30$ in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.

2004 Paraguay Mathematical Olympiad, 5

Tags: number theory , perfect square , digit

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 $A$.

1972 All Soviet Union Mathematical Olympiad, 161

Tags: perfect square , maximum , number theory

Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.

1926 Eotvos Mathematical Competition, 2

Tags: consecutive , perfect square , number theory

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

2023 China Northern MO, 6

Tags: perfect square , number theory

A positive integer $m$ is called a [i]beautiful [/i] integer if that there exists a positive integer $n$ such that $m = n^2+ n + 1$. Prove that there are infinitely many [i]beautiful [/i] integers with square factors, and the square factors of different beautiful integers are relative prime.

2006 All-Russian Olympiad Regional Round, 11.7

Tags: perfect square , sum of squares , number theory

Prove that if a natural number $N$ is represented in the form as the sum of three squares of integers divisible by $3$, then it is also represented as the sum of three squares of integers not divisible by $3$.