MathDB

2

Part of 2015 Iran MO (2nd Round)

Problems(2)

Iran Second Round 2015 - Problem 2 day 1

Source:

5/8/2015
There's a special computer and it has a memory. At first, it's memory just contains xx. We fill up the memory with the following rules.
1) If f≠0f\neq 0 is in the memory, then we can also put 1f\frac{1}{f} in it.
2) If f,gf,g are in the memory, then we can also put f+g f+g and f−gf-g in it.
Find all natural number nn such that we can have xnx^n in the memory.
algebra
Circle and Arcs

Source: Iran Second Round 2015 - Problem 5 day 2

5/8/2015
A circle is divided into 2n2n equal by 2n2n points. Ali draws n+1n+1 arcs, of length 1,2,…,n+11,2,\ldots,n+1. Prove that we can find two arcs, such that one of them is inside in the other one.
combinatoricsIran