Olympiad problems

Estonia Open Junior - geometry, 2001.2.2

In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.

perpendicular geometry consecutive integers

II Soros Olympiad 1995 - 96 (Russia), 11.7

Find three consecutive natural numbers, each of which is divisible by the square of the sum of its digits. Prove that there are no five such numbers in a row.

summation number theory consecutive integers

2014 JHMMC 7 Contest, 14

$11$ consecutive integers sum to $1331$. What is the largest of the $11$ integers?

consecutive integers algebra

2013 India PRMO, 18

What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?

consecutive integers sum maximum algebra number theory

2009 Puerto Rico Team Selection Test, 2

In each box of a $ 1 \times 2009$ grid, we place either a $ 0$ or a $ 1$, such that the sum of any $ 90$ consecutive boxes is $ 65$. Determine all possible values of the sum of the $ 2009$ boxes in the grid.

combinatorics consecutive integers sum

2017 Hanoi Open Mathematics Competitions, 12

Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?

prime number theory consecutive integers prime numbers sequences