Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2005 Indonesia MO Problems

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 $n$.

Solution

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 $11p(n)=n^2-2005$.

Solution

Problem 3

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

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

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

Solution

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 $\triangle ABC$.

Solution

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 $m-\left\lfloor \frac{m}{2005}\right\rfloor=2005$.

Solution

Problem 6

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

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

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

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

Solution

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 $LN$.

Solution

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.

Solution

See Also

2005 Indonesia MO (Problems)
Preceded by
2004 Indonesia MO 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 Followed by
2006 Indonesia MO
All Indonesia MO Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2005_Indonesia_MO_Problems&oldid=97658"