Olympiad problems

2025 Philippine MO, P8

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\] is a perfect cube.

1958 Polish MO Finals, 1

Tags: perfect cube , divide , number theory

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

1999 Bundeswettbewerb Mathematik, 4

Tags: perfect square , perfect cube , 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].

1999 Greece JBMO TST, 4

Tags: perfect power , number theory , perfect cube

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

2000 All-Russian Olympiad Regional Round, 9.5

Tags: combinatorics , number theory , perfect cube

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]

2020 Canadian Mathematical Olympiad Qualification, 5

Tags: sequence , perfect cube , number theory , number theory with sequences

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.

2005 Bosnia and Herzegovina Team Selection Test, 6

Tags: integer , number theory , perfect cube

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.

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

1997 IMO Shortlist, 15

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.

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 $(n + 1)!$

2002 Brazil National Olympiad, 1

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

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.

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.

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 $L(C ) = 1996$.

1967 IMO Longlists, 38

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.

1909 Eotvos Mathematical Competition, 1

Tags: perfect cube , number theory , consecutive

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

2017 NZMOC Camp Selection Problems, 3

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

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

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.

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.

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.