# 2012 Indonesia MO Problems

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## Day 1

### Problem 1

Show that for any positive integers $a$ and $b$, the number $$n=\mathrm{LCM}(a,b)+\mathrm{GCD}(a,b)-a-b$$ is an even non-negative integer.

### Problem 2

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that $$(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.$$

### Problem 3

Given an acute triangle $ABC$ with $AB>AC$ that has circumcenter $O$. Line $BO$ and $CO$ meet the bisector of $\angle BAC$ at $P$ and $Q$, respectively. Moreover, line $BQ$ and $CP$ meet at $R$. Show that $AR$ is perpendicular to $BC$.

### Problem 4

Given $2012$ distinct points $A_1,A_2,\dots,A_{2012}$ on the Cartesian plane. For any permutation $B_1,B_2,\dots,B_{2012}$ of $A_1,A_2,\dots,A_{2012}$ define the shadow of a point $P$ as follows: Point $P$ is rotated by $180^{\circ}$ around $B_1$ resulting $P_1$, point $P_1$ is rotated by $180^{\circ}$ around $B_2$ resulting $P_2$, ..., point $P_{2011}$ is rotated by $180^{\circ}$ around $B_{2012}$ resulting $P_{2012}$. Then, $P_{2012}$ is called the shadow of $P$ with respect to the permutation $B_1,B_2,\dots,B_{2012}$. Let $N$ be the number of different shadows of $P$ up to all permutations of $A_1,A_2,\dots,A_{2012}$. Determine the maximum value of $N$.

## Day 2

### Problem 5

Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is $$\left[ \begin{array}{cccc} 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right].$$ Let those two collections satisfy the following properties: (i) On each row of $P$, from left to right, the numbers are non-increasing, (ii) On each column of $Q$, from top to bottom, the numbers are non-increasing, (iii) The sum of numbers on the row in $P$ equals to the same row in $Q$, (iv) The sum of numbers on the column in $P$ equals to the same column in $Q$. Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$.

### Problem 6

Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying $$f(x+y)=f(x)+f(y)+\dfrac{1}{2012}$$ for all positive real numbers $x$ and $y$.

### Problem 7

Let $n$ be a positive integer. Show that the equation $$\sqrt{x}+\sqrt{y}=\sqrt{n}$$ have solution of pairs of positive integers $(x,y)$ if and only if $n$ is divisible by some perfect square greater than $1$.

### Problem 8

Given a triangle $ABC$, let the bisector of $\angle BAC$ meets the side $BC$ and circumcircle of triangle $ABC$ at $D$ and $E$, respectively. Let $M$ and $N$ be the midpoints of $BD$ and $CE$, respectively. Circumcircle of triangle $ABD$ meets $AN$ at $Q$. Circle passing through $A$ that is tangent to $BC$ at $D$ meets line $AM$ and side $AC$ respectively at $P$ and $R$. Show that the four points $B,P,Q,R$ lie on the same line.