Olympiad problems

2024 Czech-Polish-Slovak Junior Match, 2

Tags: geometry , product , perfect cube , 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?

2009 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: perfect cube , perfect square , number theory

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

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.

2017 NZMOC Camp Selection Problems, 3

Tags: perfect cube , 3d geometry , number theory , geometry

Find all prime numbers $p$ such that $16p + 1$ is a perfect cube.

1992 ITAMO, 6

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.

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.

2018 Irish Math Olympiad, 9

Tags: recurrence relation , perfect cube , sequence , number theory

The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$. Prove that there exists at most one $n$ for which $a_n$ is the cube of 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 $(n + 1)!$

2014 Thailand Mathematical Olympiad, 6

Tags: number theory , perfect cube

Find all primes $p$ such that $2p^2 - 3p - 1$ is a positive perfect cube

1979 IMO Longlists, 69

Tags: counting , number theory , perfect cube , perfect square , 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 $N >M.$

1967 Dutch Mathematical Olympiad, 2

Tags: arithmetic sequence , number theory , perfect cube

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.

1909 Eotvos Mathematical Competition, 1

Tags: number theory , consecutive , perfect cube

Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

2008 JBMO Shortlist, 10

Tags: number theory , perfect cube

Prove that $2^n + 3^n$ is not a perfect cube for any positive integer $n$.

2013 Thailand Mathematical Olympiad, 10

Tags: prime , perfect cube , number theory

Find all pairs of positive integers $(x, y)$ such that $\frac{xy^3}{x+y}$ is the cube of a prime.

1969 IMO Longlists, 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.

2002 Brazil National Olympiad, 1

Tags: number theory , subset , perfect cube , perfect power , sum , perfect square

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

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 $n$ .

1951 Moscow Mathematical Olympiad, 197

Tags: geometry , number theory , perfect cube , 3d geometry

Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.

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.

2020 Azerbaijan Senior NMO, 2

Tags: number theory , perfect cube , diophantine equation

$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$

2017 Argentina National Olympiad, 4

Tags: number theory , perfect square , perfect cube

For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$ Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .

2002 Paraguay Mathematical Olympiad, 4

Tags: number theory , perfect cube

Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.

2017 Ecuador Juniors, 6

Tags: number theory , perfect cube

Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.

2011 May Olympiad, 2

Tags: perfect square , digit , perfect cube , number theory

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.

1959 Poland - Second Round, 4

Tags: perfect cube , number theory , divide

Given a sequence of numbers $ 13, 25, 43, \ldots $ whose $ n $-th term is defined by the formula $$a_n =3(n^2 + n) + 7$$ Prove that this sequence has the following properties: 1) Of every five consecutive terms of the sequence, exactly one is divisible by $ 5 $, 2( No term of the sequence is the cube of an integer.