2004 Indonesia MO Problems

Day 1

Problem 1

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

Solution

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?

Solution

Problem 3

In how many ways can we change the sign $\ast$ with $+$ or $-$, such that the following equation is true?

\[1 \ast 2 \ast 3 \ast 4 \ast 5 \ast 6 \ast 7 \ast 8 \ast 9 \ast 10 = 29\]

Solution

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.

Solution

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 $S = 16x_1 + 25x_2 + 36x_3 + 49x_4 + 64x_5 + 81x_6 + 100x_7$.

Solution

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.

Solution

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 $a + b - c$.

Solution

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 $1/5 \text{ m}^2$.

Solution

See Also

2004 Indonesia MO (Problems)
Preceded by
2003 Indonesia MO
1 2 3 4 5 6 7 8 Followed by
2005 Indonesia MO
All Indonesia MO Problems and Solutions
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