This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 76

1997 Tournament Of Towns, (533) 5
Tags: number theory , perfect cube , consecutive
Prove that the number (a) $97^{97}$ (b) $1997^{17}$ cannot be equal to a sum of cubes of several consecutive integers. (AA Egorov)

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]

2011 May Olympiad, 2
Tags: perfect square , number theory , digit , perfect cube
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.

1996 Estonia Team Selection Test, 1
Tags: number theory , divisibility , perfect cube
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.

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.

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.

2021 Bangladeshi National Mathematical Olympiad, 7
Tags: number theory , function , perfect cube , divisor , perfect square
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 $100$.

2005 Bosnia and Herzegovina Team Selection Test, 6
Tags: number theory , perfect cube , integer
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.

2022 Assara - South Russian Girl's MO, 2
Tags: number theory , perfect cube
There are $2022$ natural numbers written in a row. Product of any two adjacent numbers is a perfect cube. Prove that the product of the two extremes is also a perfect cube.

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.

2012 Belarus Team Selection Test, 1
Tags: number theory , perfect cube
For $n$ positive integers $a_1,...,a_n$ consider all their pairwise products $a_ia_j$, $1 \le i < j \le n$. Let $N$ be the number of those products which are the cubes of positive integers. Find the maximal possible value of $N$ if it is known that none of $a_j$ is a cube of an integer. (S. Mazanik)

1967 IMO Shortlist, 1
Tags: algebra , number theory , decimal representation , perfect cube
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

2016 Hanoi Open Mathematics Competitions, 8
Tags: number theory , sum , sum of digits , perfect cube
Determine all $3$-digit numbers which are equal to cube of the sum of all its digits.

1969 IMO Shortlist, 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.

2017 Purple Comet Problems, 14
Tags: number theory , perfect cube
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.

II Soros Olympiad 1995 - 96 (Russia), 11.6
Tags: number theory , perfect cube
For what natural number $x$ will the value of the polynomial $x^3+7x^2+6x+1$ be the cube of a natural number?

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.

1909 Eotvos Mathematical Competition, 1
Tags: number theory , consecutive , perfect cube
Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

1999 Bundeswettbewerb Mathematik, 4
Tags: number theory , perfect square , perfect cube
A natural number is called [i]bright [/i] if it is the sum of a perfect square and a perfect cube. Prove that if $r$ and $s$ are any two positive integers, then (a) there exist infinitely many positive integers $n$ such that both $r+n$ and $s+n$ are [i]bright[/i], (b) there exist infinitely many positive integers $m$ such that both rm and sm are [i]bright[/i].

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.

2025 Philippine MO, P8
Tags: number theory , function , perfect cube
Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.

2020 Azerbaijan Senior NMO, 2
Tags: perfect cube , diophantine equation , number theory
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

2017 NZMOC Camp Selection Problems, 3
Tags: perfect cube , number theory , 3d geometry , geometry
Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

1998 Romania Team Selection Test, 2
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.