Olympiad problems

2023 4th Memorial "Aleksandar Blazhevski-Cane", P1

Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. [i]Proposed by Ilir Snopce[/i]

2022 Cyprus JBMO TST, 1

Tags: Perfect Square , number theory

Find all integer values of $x$ for which the value of the expression \[x^2+6x+33\] is a perfect square.

2011 Denmark MO - Mohr Contest, 1

Tags: number theory , Perfect Square , combinatorics

Georg writes the numbers from $1$ to $15$ on different pieces of paper. He attempts to sort these pieces of paper into two stacks so that none of the stacks contains two numbers whose sum is a square number.Prove that this is impossible. (The square numbers are the numbers $0 = 0^2$, $1 = 1^2$, $4 = 2^2$, $9 = 3^2$ etc.)

1989 Swedish Mathematical Competition, 3

Tags: Perfect Square , number theory , geometry , 3D geometry

Find all positive integers $n$ such that $n^3 - 18n^2 + 115n - 391$ is the cube of a positive intege

1997 Tuymaada Olympiad, 1

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

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

2018 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: sum of cubes , Perfect Square , perfect cube , number theory , Sum of Squares

Determine if there are $2018$ different positive integers such that the sum of their squares is a perfect cube and the sum of their cubes is a perfect square.

2023 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , Perfect Squares , Perfect Square , sum of digits

Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.

2001 Romania Team Selection Test, 4

Tags: number theory , Additive Number Theory , Sum of Squares , Perfect Square , IMO Shortlist

Show that the set of positive integers that cannot be represented as a sum of distinct perfect squares is finite.

2020 Dutch Mathematical Olympiad, 4

Tags: number theory , Perfect Square , prime

Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.

2019 Girls in Mathematics Tournament, 1

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

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.

2008 Postal Coaching, 2

Tags: polynomial , Integer Polynomial , Perfect Square , algebra

Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.

2014 Taiwan TST Round 3, 2

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.

1997 Tournament Of Towns, (562) 3

Tags: number theory , Integer , Perfect Square

All expressions of the form $$\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}$$ (with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer. (A Kanel)

2020 Thailand TST, 3

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]

2000 Abels Math Contest (Norwegian MO), 1b

Tags: Perfect Square , number theory

Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square

2008 Postal Coaching, 4

Tags: number theory , Perfect Square , perfect cube

Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.

2006 All-Russian Olympiad Regional Round, 9.8

Tags: number theory , Sum of Squares , Perfect Square

A number $N$ that is not divisible by $81$ can be represented as a sum of squares of three integers divisible by $3$. Prove that it is also representable as the sum of the squares of three integers not divisible by $3$.

1966 Swedish Mathematical Competition, 3

Tags: Sum of Squares , number theory , Perfect Square

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

1988 Austrian-Polish Competition, 5

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

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.