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Problems of the IMO 2005 Merida, Mexico

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Home » Forum » Olympiad Section » High-School Contests » International Mathematical Olympiad » Previous IMOs » IMO 2005 Mexico » Questions of IMO 2005 Mexico

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Valentin Vornicu
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Problems of the IMO 2005 Merida, Mexico
All the 6 problems - no solution in this topic!!

$\textrm{ 46th International Mathematical Olympiad} \\ \textrm{ July 13, 2005 - Day I}$ Problem 1
Six points are chosen on the sides of an equilateral triangle $ABC$ : $A_1$ , $A_2$ on $BC$ , $B_1$ , $B_2$ on $CA$ and $C_1$ , $C_2$ on $AB$ , such that they are the vertices of a convex hexagon $A_1A_2B_1B_2C_1C_2$ with equal side lengths.

Prove that the lines $A_1B_2$ , $B_1C_2$ and $C_1A_2$ are concurrent.

Problem 2
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$ .

Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$ .

Problem 3
Let $x,y,z$ be three positive reals such that $xyz\geq 1$ . Prove that
$\frac { x^5-x^2 }{x^5+y^2+z^2} + \frac {y^5-y^2}{x^2+y^5+z^2} + \frac {z^5-z^2}{x^2+y^2+z^5} \geq 0 .$
$\textrm{ July 14, 2005 - Day II}$
Problem 4
Determine all positive integers relatively prime to all the terms of the infinite sequence $a_n=2^n+3^n+6^n -1$ , $n\geq 1$ .

Problem 5
Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$ . Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$ , respectively and satisfy $BE=DF$ . The lines $AC$ and $BD$ meet at $P$ , the lines $BD$ and $EF$ meet at $Q$ , the lines $EF$ and $AC$ meet at $R$ .

Prove that the circumcircles of the triangles $PQR$ , as $E$ and $F$ vary, have a common point other than $P$ .

Problem 6
In a mathematical competition in which 6 problems were posed to be participants, every two of these problems were solved by more than $\dfrac 25$ of the contestants. Moreover, no contestant solved all the 6 problems. Show that there are at least 2 contestants who solved exactly 5 problems each.

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Posted: Wed Jul 13, 2005 8:03 pm

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