Olympiad problems

1996 Estonia Team Selection Test, 1

Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.

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.

1967 IMO Shortlist, 1

Tags: algebra , decimal representation , perfect cube , number theory

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

II Soros Olympiad 1995 - 96 (Russia), 11.6

Tags: perfect cube , number theory

For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?

2021 Bangladeshi National Mathematical Olympiad, 7

Tags: perfect square , number theory , function , perfect cube , divisor

For a positive integer $n$, let $s(n)$ and $c(n)$ be the number of divisors of $n$ that are perfect squares and perfect cubes respectively. A positive integer $n$ is called fair if $s(n)=c(n)>1$. Find the number of fair integers less than $100$.

2010 Saudi Arabia IMO TST, 3

Tags: perfect cube , geometry , 3d geometry , number theory

Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

2012 Ukraine Team Selection Test, 3

Tags: perfect cube , sum of cubes , number theory , perfect number

A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.

1998 Singapore Team Selection Test, 3

Tags: number theory , modular arithmetic , perfect square , perfect cube , arithmetic sequence

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

1992 ITAMO, 6

Tags: perfect cube , rational , number theory

Let $a$ and $b$ be integers. Prove that if $\sqrt[3]{a}+\sqrt[3]{b}$ is a rational number, then both $a$ and $b$ are perfect cubes.

2012 Estonia Team Selection Test, 1

Tags: perfect square , perfect cube , number theory

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

2018 Irish Math Olympiad, 9

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

The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$. Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.

2016 May Olympiad, 1

Tags: perfect cube , number theory

Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ cannot be smaller.

2020 Azerbaijan Senior NMO, 2

Tags: perfect cube , diophantine equation , number theory

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

2011 May Olympiad, 2

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

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.

1969 IMO Longlists, 7

Tags: modular arithmetic , number theory , diophantine equation , additive number theory , perfect cube

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

1979 IMO Longlists, 69

Tags: number theory , perfect cube , perfect square , counting , additive number theory

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 $N >M.$

2016 Stars of Mathematics, 1

Tags: number theory , perfect cube

Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $

2023 Indonesia Regional, 2

Tags: number theory , perfect cube

Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that \[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\] are all perfect cubes. (a) Prove that $K \ne 2$ and $K \ne 4$ (b) Find the minimum value of $K$ that satisfies. [i]Proposed by Muhammad Afifurrahman[/i]

2017 Argentina National Olympiad, 4

Tags: perfect square , perfect cube , number theory

For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$ Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .

1959 Poland - Second Round, 4

Tags: number theory , perfect cube , divide

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.

1951 Moscow Mathematical Olympiad, 197

Tags: perfect cube , number theory , geometry , 3d geometry

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

2013 Thailand Mathematical Olympiad, 10

Tags: prime , perfect cube , number theory

Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.

2012 Thailand Mathematical Olympiad, 12

Tags: number theory , integer , perfect cube

Let $a, b, c$ be positive integers. Show that if $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer then $abc$ is a perfect cube.

1979 IMO Shortlist, 21

Tags: number theory , perfect cube , perfect square , counting , additive number theory

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 $N >M.$

2024 Czech-Polish-Slovak Junior Match, 2

Tags: product , perfect cube , geometry , 3d geometry , number theory

How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?