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Difference between revisions of "2004 IMO Problems"

(Created page with "Problems of the 46th IMO 2004 Mérida, Mexico. == Day 1 == === Problem 1 === Six points are chosen on the sides of an equilateral triangle <math>ABC</math>: <math>A_1, A...")
 
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Problems of the 46th [[IMO]] 2004 Mérida, Mexico.
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Problems of the 45th [[IMO]] 2004 Athens, Greece.
  
 
== Day 1 ==
 
== Day 1 ==
  
 
=== Problem 1 ===
 
=== Problem 1 ===
−
Six points are chosen on the sides of an equilateral triangle <math>ABC</math>: <math>A_1, A_2</math> on <math>BC</math>, <math>B_1</math>, <math>B_2</math> on <math>CA</math> and <math>C_1</math>, <math>C_2</math> on <math>AB</math>, such that they are the vertices of a convex hexagon <math>A_1A_2B_1B_2C_1C_2</math> with equal side lengths. Prove that the lines <math>A_1B_2, B_1C_2</math> and <math>C_1A_2</math> are concurrent.
+
Let <math>ABC</math> be an acute-angled triangle with <math>AB\neq AC</math>. The circle with diameter <math>BC</math> intersects the sides <math>AB</math> and <math>AC</math> at <math>M</math> and <math>N</math> respectively. Denote by <math>O</math> the midpoint of the side <math>BC</math>. The bisectors of the angles <math>\angle BAC</math> and <math>\angle MON</math> intersect at <math>R</math>. Prove that the circumcircles of the triangles <math>BMR</math> and <math>CNR</math> have a common point lying on the side <math>BC</math>.
  
 
[[2004 IMO Problems/Problem 1 | Solution]]
 
[[2004 IMO Problems/Problem 1 | Solution]]
  
 
=== Problem 2 ===
 
=== Problem 2 ===
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Let <math>a_1, a_2, \dots</math> be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer <math>n</math> the numbers <math>a_1, a_2, \dots, a_n</math> leave <math>n</math> different remainders upon division by <math>n</math>. Prove that every integer occurs exactly once in the sequence.
+
Find all polynomials <math>f</math> with real coefficients such that for all reals <math>a,b,c</math> such that <math>ab + bc + ca = 0</math> we have the following relations
 +
 
 +
<cmath>f(a - b) + f(b - c) + f(c - a) = 2f(a + b + c).</cmath>
  
 
[[2004 IMO Problems/Problem 2 | Solution]]
 
[[2004 IMO Problems/Problem 2 | Solution]]
  
 
=== Problem 3 ===
 
=== Problem 3 ===
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Let <math>x, y, z > 0</math> satisfy <math>xyz\ge 1</math>. Prove that<cmath>\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} \ge 0.</cmath>
+
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
 +
 
 +
<asy>
 +
unitsize(0.5 cm);
 +
 
 +
draw((0,0)--(1,0));
 +
draw((0,1)--(1,1));
 +
draw((2,1)--(3,1));
 +
draw((0,2)--(3,2));
 +
draw((0,3)--(3,3));
 +
draw((0,0)--(0,3));
 +
draw((1,0)--(1,3));
 +
draw((2,1)--(2,3));
 +
draw((3,1)--(3,3));
 +
</asy>
 +
 
 +
Determine all <math>m \times n</math> rectangles that can be covered without gaps and without overlaps with hooks such that;
 +
(a) the rectangle is covered without gaps and without overlaps,
 +
(b) no part of a hook covers area outside the rectangle.
  
 
[[2004 IMO Problems/Problem 3 | Solution]]
 
[[2004 IMO Problems/Problem 3 | Solution]]
Line 21: Line 41:
  
 
=== Problem 4 ===
 
=== Problem 4 ===
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Determine all positive integers relatively prime to all the terms of the infinite sequence<cmath>a_n=2^n+3^n+6^n -1,\ n\geq 1.</cmath>
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Let <math>n \geq 3</math> be an integer. Let <math>t_1</math>, <math>t_2</math>, ..., <math>t_n</math> be positive real numbers such that
 +
 
 +
<cmath>n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right).</cmath>
 +
Show that <math>t_i</math>, <math>t_j</math>, <math>t_k</math> are side lengths of a triangle for all <math>i</math>, <math>j</math>, <math>k</math> with <math>1 \leq i < j < k \leq n</math>.
  
 
[[2004 IMO Problems/Problem 4 | Solution]]
 
[[2004 IMO Problems/Problem 4 | Solution]]
  
 
=== Problem 5 ===
 
=== Problem 5 ===
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Let <math>ABCD</math> be a fixed convex quadrilateral with <math>BC = DA</math> and <math>BC \nparallel DA</math>. Let two variable points <math>E</math> and <math>F</math> lie of the sides <math>BC</math> and <math>DA</math>, respectively, and satisfy <math>BE = DF</math>. The lines <math>AC</math> and <math>BD</math> meet at <math>P</math>, the lines <math>BD</math> and <math>EF</math> meet at <math>Q</math>, the lines <math>EF</math> and <math>AC</math> meet at <math>R</math>. Prove that the circumcircles of the triangles <math>PQR</math>, as <math>E</math> and <math>F</math> vary, have a common point other than <math>P</math>.
+
In a convex quadrilateral <math>ABCD</math>, the diagonal <math>BD</math> bisects neither the angle <math>ABC</math> nor the angle <math>CDA</math>. The point <math>P</math> lies inside <math>ABCD</math> and satisfies<cmath>\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.</cmath>
 +
Prove that <math>ABCD</math> is a cyclic quadrilateral if and only if <math>AP = CP.</math>
  
 
[[2004 IMO Problems/Problem 5 | Solution]]
 
[[2004 IMO Problems/Problem 5 | Solution]]
  
 
=== Problem 6 ===
 
=== Problem 6 ===
−
In a mathematical competition, in which 6 problems were posed to the participants, every two of these problems were solved by more than 2/5 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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We call a positive integer alternating if every two consecutive digits in its decimal representation have a different parity.
 +
 
 +
Find all positive integers <math>n</math> such that <math>n</math> has a multiple which is alternating.
  
 
[[2004 IMO Problems/Problem 6 | Solution]]
 
[[2004 IMO Problems/Problem 6 | Solution]]

Revision as of 23:53, 20 February 2021

Problems of the 45th IMO 2004 Athens, Greece.

Day 1

Problem 1

Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

Solution

Problem 2

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab + bc + ca = 0$ we have the following relations

\[f(a - b) + f(b - c) + f(c - a) = 2f(a + b + c).\]

Solution

Problem 3

Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.

[asy] unitsize(0.5 cm); draw((0,0)--(1,0)); draw((0,1)--(1,1)); draw((2,1)--(3,1)); draw((0,2)--(3,2)); draw((0,3)--(3,3)); draw((0,0)--(0,3)); draw((1,0)--(1,3)); draw((2,1)--(2,3)); draw((3,1)--(3,3)); [/asy]

Determine all $m \times n$ rectangles that can be covered without gaps and without overlaps with hooks such that; (a) the rectangle is covered without gaps and without overlaps, (b) no part of a hook covers area outside the rectangle.

Solution

Day 2

Problem 4

Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that

\[n^2 + 1 > \left( t_1 + t_2 + ... + t_n \right) \left( \frac {1}{t_1} + \frac {1}{t_2} + ... + \frac {1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.

Solution

Problem 5

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies\[\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP.$

Solution

Problem 6

We call a positive integer alternating if every two consecutive digits in its decimal representation have a different parity.

Find all positive integers $n$ such that $n$ has a multiple which is alternating.

Solution

Resources

2004 IMO (Problems) • Resources
Preceded by
2003 IMO Problems 		1 • 2 • 3 • 4 • 5 • 6 Followed by
2005 IMO Problems
All IMO Problems and Solutions
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