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

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.

1967 Dutch Mathematical Olympiad, 2
Tags: perfect cube , number theory , arithmetic sequence
Consider arithmetic sequences where all terms are natural numbers. If the first term of such a sequence is $1$, prove that that sequence contains infinitely many terms that are the cube of a natural number. Give an example of such a sequence in which no term is the cube of a natural number and show the correctness of this example.

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

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)

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

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

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.

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.

1980 Bundeswettbewerb Mathematik, 1
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.

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.

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

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

2017 Ecuador Juniors, 6
Tags: number theory , perfect cube
Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

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.

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?

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.

1985 Poland - Second Round, 4
Tags: algebra , rational , radical , perfect cube , 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.

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

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 ?

2023 Regional Olympiad of Mexico West, 4
Tags: number theory , perfect square , perfect cube , perfect power
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.

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

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)

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.

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.

2013 IMAC Arhimede, 2
Tags: number theory , perfect cube , divisible , exponential
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.