Olympiad problems

2002 Brazil National Olympiad, 1

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.

1959 Poland - Second Round, 4

Tags: number theory , perfect cube , 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.

1967 IMO Longlists, 1

Tags: algebra , number theory , decimal representation , perfect cube

Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.

2005 Argentina National Olympiad, 4

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

We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ . Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners. Prove that Germán can always achieve his goal.

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.

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

1992 ITAMO, 6

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.

1958 Polish MO Finals, 1

Tags: number theory , perfect cube , divide

Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

2016 May Olympiad, 1

Tags: number theory , perfect cube

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.

2018 Irish Math Olympiad, 9

Tags: perfect cube , sequence , recurrence relation , 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.

2009 Abels Math Contest (Norwegian MO) Final, 1b

Tags: number theory , perfect cube , perfect square

Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.

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.

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.

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.

2008 Postal Coaching, 4

Tags: number theory , perfect square , perfect cube

Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.

1967 Dutch Mathematical Olympiad, 2

Tags: perfect cube , arithmetic sequence , number theory

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.

2017 Argentina National Olympiad, 4

Tags: perfect square , perfect cube , number theory

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 $1^3,2^3,3^3,4^3,…$ .

2017 Czech-Polish-Slovak Junior Match, 1

Tags: perfect cube , number theory , prime

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.

2020 Canadian Mathematical Olympiad Qualification, 5

Tags: sequence , perfect cube , 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.

1967 IMO Shortlist, 4

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.

1949-56 Chisinau City MO, 3

Tags: perfect cube , geometry , 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.

2000 All-Russian Olympiad Regional Round, 9.5

Tags: combinatorics , perfect cube , number theory

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: 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.

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.

2017 Ecuador Juniors, 6

Tags: number theory , perfect cube

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