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Difference between revisions of "2003 Indonesia MO Problems"

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===Problem 8===
 
===Problem 8===
  
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Given a triangle <math>ABC</math> with <math>C</math> as the right angle, and the sidelengths of the triangle are integers.  Determine the sidelengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.
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Given a triangle <math>ABC</math> with <math>C</math> as the right angle, and the sidelengths of the triangle are integers.  Determine the possible side lengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.
  
 
[[2003 Indonesia MO Problems/Problem 8|Solution]]
 
[[2003 Indonesia MO Problems/Problem 8|Solution]]

Latest revision as of 01:20, 19 May 2024

Day 1

Problem 1

Prove that $a^9 - a$ is divisible by $6$ for every integers $a$.

Solution

Problem 2

Given a quadrilateral $ABCD$. Let $P$, $Q$, $R$, and $S$ are the midpoints of $AB$, $BC$, $CD$, and $DA$, respectively. $PR$ and $QS$ intersects at $O$. Prove that $PO = OR$ and $QO = OS$.

Solution

Problem 3

Find all real solutions of the equation $\lfloor x^2 \rfloor + \lceil x^2 \rceil = 2003$.

[Note: For any real number $\alpha$, $\lfloor \alpha \rfloor$ is the largest integer less than or equal to $\alpha$, and $\lceil \alpha \rceil$ denote the smallest integer more than or equal to $\alpha$.]

Solution

Problem 4

Given a $19 \times 19$ matrix, where each element is valued $+1$ or $-1$. Let $b_i$ be the product of all elements at the $i^\text{th}$ row, and $k_j$ be the product of all elements at the $j^\text{th}$ column. Prove that:

\[b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \ne 0\]

Solution

Day 2

Problem 5

For every real number $a,b,c$, prove the following inequality

\[5a^2 + 5b^2 + 5c^2 \ge 4ab + 4bc + 4ca\]

and determine when the equality holds.

Solution

Problem 6

A hall of a palace is in a shape of regular hexagon, where the sidelength is $6 \text{ m}$. The floor of the hall is covered with an equilateral triangle-shaped tile with sidelength $50 \text{ cm}$. Every tile is divided into $3$ congruent triangles (refer to the figure). Every triangle-region is colored with a certain color so that each tile has $3$ different colors. The King wants to ensure that no two tiles have the same color pattern. At least, how many colors are needed?

[asy] pair C=(25,14.434); draw((0,0)--(50,0)--(25,43.301)--(0,0)); draw(C--(0,0)); draw(C--(50,0)); draw(C--(25,43.301)); [/asy]

Solution

Problem 7

Let $k,m,n$ be positive integers such that $k > n > 1$ and the greatest common divisor of $k$ and $n$ is $1$. Prove that if $k-n$ divides $k^m - n^{m-1}$, then $k \le 2n-1$.

Solution

Problem 8

Given a triangle $ABC$ with $C$ as the right angle, and the sidelengths of the triangle are integers. Determine the possible side lengths of the triangle if the product of the legs of the right triangle equals to three times the perimeter of the triangle.

Solution

See Also

2003 Indonesia MO (Problems)
Preceded by
2002 Indonesia MO 		1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 Followed by
2004 Indonesia MO
All Indonesia MO Problems and Solutions
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