# 2004 Indonesia MO Problems

## Day 1

### Problem 1

How many odd and even divisors of $5^6 - 1$ are there?

### Problem 2

A trough, if filled with cold water tap, will be full in 14 minutes. To empty the full trough with opening the hole on the base of the trough, the water will be all out in 21 minutes. If the cold water tap and the hot water tap are opened simultaneously with the opening of the hole, the trough will be full in 12.6 minutes. Then, how long does it take to full the trough when only the hot water tap is opened and the hole is closed?

### Problem 3

In how many ways can we change the sign $\ast$ with $+$ or $-$, such that the following equation is true? $$1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29$$

### Problem 4

There exists 4 circles, $a,b,c,d$, such that $a$ is tangent to both $b$ and $d$, $b$ is tangent to both $a$ and $c$, $c$ is both tangent to $b$ and $d$, and $d$ is both tangent to $a$ and $c$. Show that all these tangent points are located on a circle.

## Day 2

### Problem 5

Given a system of equations: $\left\{\begin{array}{l}x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 = 1\\4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 = 12\\9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 = 123\end{array}\right.$

Then determine the value of $S = 16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7$.

### Problem 6

A quadratic equation $x^2 + ax + b + 1 = 0$ with integers $a$ and $b$ has roots which are positive integers. Prove that $a^2 + b^2$ is not a prime.

### Problem 7

Prove that in a triangle $ABC$ with $C$ as the right angle, where $a$ denote the side in front of angle $A$, $b$ denote the side in front of angle $B$, $c$ denote the side in front of angle $C$, the diameter of the incircle of $ABC$ equals to $a + b - c$.

### Problem 8

A floor with an area of $3 \text{ m}^2$ will be covered by $5$ rugs with various shapes, each having an area of $1 \text{ m}^2$. Show that there exist $2$ overlapping rugs with the overlapped area at least $1/5 \text{ m}^2$.