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: 9

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

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

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

2016 May Olympiad, 1
Tags: number theory , perfect cubes
Seven different positive integers are written on a sheet of paper. The result of the multiplication of the seven numbers is the cube of a whole number. If the largest of the numbers written on the sheet is $N$, determine the smallest possible value of $N$. Show an example for that value of $N$ and explain why $N$ cannot be smaller.

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

2023 Indonesia Regional, 2
Tags: number theory , perfect cubes , Indonesia , RMO
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]

1998 Chile National Olympiad, 5
Tags: perfect cubes , 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.

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

2011 May Olympiad, 2
Tags: Perfect Square , perfect cubes , number theory , Digits
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.