Olympiad problems

2016 Portugal MO, 2

In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?

2008 Tournament Of Towns, 1

Tags: number theory , consecutive , perfect square

An integer $N$ is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if $N > 12$

2010 Dutch Mathematical Olympiad, 2

Tags: number theory , power of 2 , consecutive

A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.

2009 All-Russian Olympiad Regional Round, 10.2

Tags: number theory , consecutive , product

Prove that there is a natural number $n > 1$ such that the product of some $n$ consecutive natural numbers is equal to the product of some $n + 100$ consecutive natural numbers.

2011 Dutch Mathematical Olympiad, 3

Tags: combinatorics , number theory , consecutive

In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each. Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$? If so, determine all values of $a$ for which this is possible.

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.

2011 Peru MO (ONEM), 1

Tags: number theory , multiple , consecutive

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.