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

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

2002 Paraguay Mathematical Olympiad, 4
Tags: perfect cube , number theory
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.

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.

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.

2010 Saudi Arabia IMO TST, 3
Tags: perfect cube , number theory , geometry , 3d geometry
Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

2009 Bosnia And Herzegovina - Regional Olympiad, 1
Tags: number theory , perfect cube , perfect square
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers

2018 BAMO, 4
Tags: number theory , perfect cube , integer , geometry , 3d geometry , algebra
(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.)

1998 Belarus Team Selection Test, 3
Tags: number theory , perfect square , perfect cube
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 ?

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.

2006 Chile National Olympiad, 4
Tags: number theory , perfect square , perfect cube
Let $n$ be a $6$-digit number, perfect square and perfect cube, if $n -6$ is neither even nor multiple of $3$. Find $n$ .

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?

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 $N >M.$

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 $N >M.$

2019 AMC 10, 11
Tags: perfect square , perfect cube
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

1996 Mexico National Olympiad, 5
Tags: combinatorics , square , perfect cube , sum
The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.

1998 Chile National Olympiad, 5
Tags: number theory , perfect cube
Show that the number $3$ can be written in a infinite number of different ways as the sum of the cubes of four integers.

1999 Greece JBMO TST, 4
Tags: number theory , perfect cube , perfect power
Examine whether exists $n \in N^*$, such that: (a) $3n$ is perfect cube, $4n$ is perfect fourth power and $5n$ perfect fifth power (b) $3n$ is perfect cube, $4n$ is perfect fourth power, $5n$ perfect fifth power and $6n$ perfect sixth power

2006 Spain Mathematical Olympiad, 2
Tags: number theory , product , consecutive , perfect square , perfect cube
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

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.

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

2006 QEDMO 2nd, 8
Tags: number theory , factorial , perfect cube
Show that for any positive integer $n\ge 4$, there exists a multiple of $n^3$ between $n!$ and $(n + 1)!$

1997 IMO Shortlist, 15
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.

2008 JBMO Shortlist, 10
Tags: perfect cube , number theory
Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.

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.

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