# 2005 Indonesia MO Problems

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## Day 1

### Problem 1

Let $n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $n$.

### Problem 2

For an arbitrary positive integer $n$, define $p(n)$ as the product of the digits of $n$ (in decimal). Find all positive integers $n$ such that $11p(n)=n^2-2005$.

### Problem 3

Let $k$ and $m$ be positive integers such that $\displaystyle\frac12\left(\sqrt{k+4\sqrt{m}}-\sqrt{k}\right)$ is an integer.

(a) Prove that $\sqrt{k}$ is rational.

(b) Prove that $\sqrt{k}$ is a positive integer.

### Problem 4

Let $M$ be a point in triangle $ABC$ such that $\angle AMC=90^{\circ}$, $\angle AMB=150^{\circ}$, $\angle BMC=120^{\circ}$. The centers of circumcircles of triangles $AMC,AMB,BMC$ are $P,Q,R$, respectively. Prove that the area of $\triangle PQR$ is greater than the area of $\triangle ABC$.

## Day 2

### Problem 5

For an arbitrary real number $x$, $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$. Prove that there is exactly one integer $m$ which satisfy $\displaystyle m-\left\lfloor \frac{m}{2005}\right\rfloor=2005$.

### Problem 6

Find all triples $(x,y,z)$ of integers which satisfy

$x(y + z) = y^2 + z^2 - 2$

$y(z + x) = z^2 + x^2 - 2$

### Problem 7

Let $ABCD$ be a convex quadrilateral. Square $AB_1A_2B$ is constructed such that the two vertices $A_2,B_1$ is located outside $ABCD$. Similarly, we construct squares $BC_1B_2C$, $CD_1C_2D$, $DA_1D_2A$. Let $K$ be the intersection of $AA_2$ and $BB_1$, $L$ be the intersection of $BB_2$ and $CC_1$, $M$ be the intersection of $CC_2$ and $DD_1$, and $N$ be the intersection of $DD_2$ and $AA_1$. Prove that $KM$ is perpendicular to $LN$.

### Problem 8

There are $90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.

$z(x + y) = x^2 + y^2 - 2$.