# 2008 Indonesia MO Problems

## Day 1

### Problem 1

Given triangle $ABC$. Points $D,E,F$ outside triangle $ABC$ are chosen such that triangles $ABD$, $BCE$, and $CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

### Problem 2

Prove that for every positive reals $x$ and $y$, $$\frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}.$$

### Problem 3

Find all positive integers which can be expressed as $$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}$$ where $a,b,c$ are positive integers that are pairwise relatively prime.

### Problem 4

Note: Problem statement slightly modified for correction.

Let $A = \{1,2,\ldots,2008\}$.

(a) Find the number of subset of $A$ such that the product of its elements is divisible by 7.

(b) Let $N(i)$ denotes the number of subset of $A$ in which the sum of its elements, when divided by 7, leaves the remainder $i$. Prove that $N(1) - N(2) + N(3) - N(4) + N(5) - N(6) = 0$.

## Day 2

### Problem 5

Let $m,n > 1$ are integers which satisfy $n|4^m - 1$ and $2^m|n - 1$. Is it a must that $n = 2^{m} + 1$?

### Problem 6

In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.

### Problem 7

Given triangle $ABC$ with sidelengths $a,b,c$. Tangents to the incircle of triangle $ABC$ that are parallel with each side of $ABC$ form three small triangles (each of them has one vertex from $A, B, C$). Prove that the sum of area of incircles of these three small triangles and the area of the incircle of triangle $ABC$ is equal to $$\frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}.$$

### Problem 8

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f: \mathbb{N}\rightarrow\mathbb{N}$ that satisfies $$f(mn)+f(m+n)=f(m)f(n)+1$$ for all natural number $m, n \in \mathbb{N}$.