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

2023 Indonesia Regional, 2
Tags: perfect cube , number theory
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]

2005 Argentina National Olympiad, 4
Tags: perfect square , 3d geometry , perfect cube , number theory , geometry
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

2021 Saudi Arabia Training Tests, 37
Tags: perfect cube , number theory
Given $n \ge 2$ distinct positive integers $a_1, a_2, ..., a_n$ none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.

1998 Singapore Team Selection Test, 3
Tags: perfect square , perfect cube , modular arithmetic , arithmetic sequence , number theory
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.

1979 IMO Shortlist, 21
Tags: perfect square , additive number theory , perfect cube , counting , 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 $N >M.$

2013 IMAC Arhimede, 2
Tags: exponential , perfect cube , number theory , divisible
For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

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

2013 Tournament of Towns, 4
Tags: sum , integer , perfect cube , number theory
Is it true that every integer is a sum of finite number of cubes of distinct integers?

2012 Thailand Mathematical Olympiad, 12
Tags: integer , perfect cube , number theory
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.

2018 BAMO, 4
Tags: perfect cube , integer , 3d geometry , algebra , number theory , geometry
(a) Find two quadruples of positive integers $(a,b, c,n)$, each with a different value of $n$ greater than $3$, such that $$\frac{a}{b} +\frac{b}{c} +\frac{c}{a} = n$$ (b) Show that if $a,b, c$ are nonzero integers such that $\frac{a}{b} +\frac{b}{c} +\frac{c}{a}$ is an integer, then $abc$ is a perfect cube. (A perfect cube is a number of the form $n^3$, where $n$ is an integer.)

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

2009 Abels Math Contest (Norwegian MO) Final, 1b
Tags: perfect cube , perfect square , number theory
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.

2001 Greece JBMO TST, 4
Tags: perfect cube , number theory
a) If positive integer $N$ is a perfect cube and is not divisible by $10$, then $N=(10m+n)^2$ where $m,n \in N$ with $1\le n\le 9$ b) Find all the positive integers $N$ which are perfect cubes, are not divisible by $10$, such that the number obtained by erasing the last three digits to be also also a perfect cube.

2015 Irish Math Olympiad, 3
Tags: perfect cube , number theory
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.

2014 NZMOC Camp Selection Problems, 3
Tags: perfect cube , number theory
Find all pairs $(x, y)$ of positive integers such that $(x + y)(x^2 + 9y)$ is the cube of a prime number.

1967 IMO Longlists, 38
Tags: modular arithmetic , perfect cube , diophantine equation , number theory
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

2025 Philippine MO, P8
Tags: function , perfect cube , number theory
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.

2016 Stars of Mathematics, 1
Tags: perfect cube , number theory
Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $

1998 Belarus Team Selection Test, 3
Tags: perfect square , perfect cube , number theory
a) Let $f(x,y) = x^3 + (3y^2+1)x^2 + (3y^4 - y^2 + 4 y - 1)x + (y^6-y^4 + 2y^3)$. Prove that if for some positive integers $a, b$ the number $f(a, b)$ is a cube of an integer then $f(a, b)$ is also a square of an integer. b) Are there infinitely many pairs of positive integers $(a, b)$ for which $f(a, b)$ is a square but not a cube ?

1967 IMO Shortlist, 4
Tags: number theory , perfect cube , diophantine equation , modular arithmetic
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

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?

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

1958 Polish MO Finals, 1
Tags: perfect cube , divide , number theory
Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

1997 IMO Shortlist, 15
Tags: perfect cube , perfect square , arithmetic sequence , number theory , modular arithmetic
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.

2008 Postal Coaching, 4
Tags: perfect square , perfect cube , number theory
Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.