Olympiad problems

2007 Bosnia and Herzegovina Junior BMO TST, 1

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

1961 Polish MO Finals, 1

Tags: number theory , consecutive

Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divide , divisible

a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

1984 Tournament Of Towns, (055) O3

Tags: square table , consecutive , combinatorics

Consider the $4(N-1)$ squares on the boundary of an $N$ by $N$ array of squares. We wish to insert in these squares $4 (N-1)$ consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number. Is this possible? Consider the cases (a) $N = 3$ (b) $N = 4$ (c) $N = 5$ (VG Boltyanskiy, Moscow)

2017 Argentina National Olympiad, 2

Tags: sum , algebra , number theory , consecutive

In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.

2018 Costa Rica - Final Round, N1

Tags: consecutive , number theory

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

2004 Tournament Of Towns, 2

Tags: combinatorics , consecutive , prime , sum

Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.

2020 Durer Math Competition Finals, 3

Tags: number theory , least common multiple , perfect square , consecutive

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

1980 Spain Mathematical Olympiad, 8

Tags: geometry , consecutive

Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.

1997 Tournament Of Towns, (533) 5

Tags: number theory , perfect cube , consecutive

Prove that the number (a) $97^{97}$ (b) $1997^{17}$ cannot be equal to a sum of cubes of several consecutive integers. (AA Egorov)