Difference between revisions of "2008 Indonesia MO Problems"

Latest revision as of 14:09, 23 June 2021

Day 1

Problem 1

Given triangle $ABC$ . Points $D,E,F$ outside triangle $ABC$ are chosen such that triangles $ABD$ , $BCE$ , and $CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Solution

Problem 2

Prove that for every positive reals $x$ and $y$ , $\frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}.$

Solution

Problem 3

Find all positive integers which can be expressed as $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}$ where $a,b,c$ are positive integers that are pairwise relatively prime.

Solution

Problem 4

Note: Problem statement slightly modified for correction.

Let $A = \{1,2,\ldots,2008\}$ .

(a) Find the number of subset of $A$ such that the product of its elements is divisible by 7.

(b) Let $N(i)$ denotes the number of subset of $A$ in which the sum of its elements, when divided by 7, leaves the remainder $i$ . Prove that $N(1) - N(2) + N(3) - N(4) + N(5) - N(6) = 0$ .

Solution

Day 2

Problem 5

Let $m,n > 1$ are integers which satisfy $n|4^m - 1$ and $2^m|n - 1$ . Is it a must that $n = 2^{m} + 1$ ?

Solution

Problem 6

In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer.

Solution

Problem 7

Given triangle $ABC$ with sidelengths $a,b,c$ . Tangents to the incircle of triangle $ABC$ that are parallel with each side of $ABC$ form three small triangles (each of them has one vertex from $A, B, C$ ). Prove that the sum of area of incircles of these three small triangles and the area of the incircle of triangle $ABC$ is equal to $\frac{\pi (a^{2}+b^{2}+c^{2})(b+c-a)(c+a-b)(a+b-c)}{(a+b+c)^{3}}.$

Solution

Problem 8

Let $\mathbb{N}$ denote the set of all positive integers. Find all functions $f: \mathbb{N}\rightarrow\mathbb{N}$ that satisfies $f(mn)+f(m+n)=f(m)f(n)+1$ for all natural number $m, n \in \mathbb{N}$ .

Solution

@@ Line 23: / Line 23: @@
 ===Problem 4===
+'''Note: Problem statement slightly modified for correction.'''
 Let <math> A = \{1,2,\ldots,2008\}</math>.
@@ Line 28: / Line 30: @@
 (a) Find the number of subset of <math> A</math> such that the product of its elements is divisible by 7.
-(b) Let <math> N(i)</math> denotes the number of subset of <math> A</math> in which the sum of its elements, when divided by 7, leaves the remainder <math> i</math>. Prove that <math> N(0) - N(1) + N(2) - N(3) + N(4) - N(5) + N(6)-N(7) = 0</math>.
+(b) Let <math> N(i)</math> denotes the number of subset of <math> A</math> in which the sum of its elements, when divided by 7, leaves the remainder <math> i</math>. Prove that <math> N(1) - N(2) + N(3) - N(4) + N(5) - N(6) = 0</math>.
 [[2008 Indonesia MO Problems/Problem 4|Solution]]

2008 Indonesia MO (Problems)
Preceded by 2007 Indonesia MO	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by 2009 Indonesia MO
All Indonesia MO Problems and Solutions

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Difference between revisions of "2008 Indonesia MO Problems"

Latest revision as of 14:09, 23 June 2021

Contents

Day 1

Problem 1

Problem 2

Problem 3

Problem 4

Day 2

Problem 5

Problem 6

Problem 7

Problem 8

See Also