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

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

2020 Canadian Mathematical Olympiad Qualification, 5
Tags: perfect cube , sequence , number theory with sequences , number theory
We define the following sequences: • Sequence $A$ has $a_n = n$. • Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise. • Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$ .• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise. • Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$ Prove that the terms of sequence E are exactly the perfect cubes.

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

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.

1999 Bundeswettbewerb Mathematik, 4
Tags: perfect cube , perfect square , number theory
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].

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

2023 Regional Olympiad of Mexico West, 4
Tags: perfect cube , perfect square , perfect power , number theory
Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

2000 All-Russian Olympiad Regional Round, 9.5
Tags: perfect cube , number theory , combinatorics
In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$. [img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]

1985 Poland - Second Round, 4
Tags: perfect cube , rational , radical , algebra , number theory
Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers.

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

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)? $\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$

2012 Ukraine Team Selection Test, 3
Tags: perfect cube , sum of cubes , perfect number , number theory
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.

2012 Belarus Team Selection Test, 1
Tags: perfect cube , number theory
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)

2022 Assara - South Russian Girl's MO, 2
Tags: perfect cube , number theory
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.

1980 Bundeswettbewerb Mathematik, 1
Tags: number theory , perfect cube , rational
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.

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.

2018 Rioplatense Mathematical Olympiad, Level 3, 1
Tags: sum of cubes , perfect square , perfect cube , sum of squares , number theory
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.

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.

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 $L(C ) = 1996$.

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

1969 IMO Shortlist, 7
Tags: modular arithmetic , diophantine equation , additive number theory , perfect cube , number theory
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

1999 Greece JBMO TST, 4
Tags: perfect cube , perfect power , number theory
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

2017 Czech-Polish-Slovak Junior Match, 1
Tags: prime , perfect cube , number theory
Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

1949-56 Chisinau City MO, 3
Tags: geometry , perfect cube , 3d geometry , number theory
Prove that the number $N = 10 ...050...01$ (1, 49 zeros, 5 , 99 zeros, 1) is a not cube of an integer.