Difference between revisions of "2008 IMO Problems"
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Problems of the 49th [[IMO]] 2008 Spain. | Problems of the 49th [[IMO]] 2008 Spain. | ||
− | == Day | + | == Day 1 == |
=== Problem 1 === | === Problem 1 === | ||
− | Let <math>H</math> be the orthocenter of an acute-angled triangle <math>ABC</math>. The circle <math>\Gamma_{A}</math> centered at the midpoint of <math>BC</math> and passing through <math>H</math> intersects | + | Let <math>H</math> be the orthocenter of an acute-angled triangle <math>ABC</math>. The circle <math>\Gamma_{A}</math> centered at the midpoint of <math>BC</math> and passing through <math>H</math> intersects line <math>BC</math> at points <math>A_{1}</math> and <math>A_{2}</math>. Similarly, define the points <math>B_{1}</math>, <math>B_{2}</math>, <math>C_{1}</math> and <math>C_{2}</math>. |
Prove that six points <math>A_{1}</math> , <math>A_{2}</math>, <math>B_{1}</math>, <math>B_{2}</math>, <math>C_{1}</math> and <math>C_{2}</math> are concyclic. | Prove that six points <math>A_{1}</math> , <math>A_{2}</math>, <math>B_{1}</math>, <math>B_{2}</math>, <math>C_{1}</math> and <math>C_{2}</math> are concyclic. | ||
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=== Problem 2 === | === Problem 2 === | ||
− | + | Let <math>x, y, z\neq 1</math> be three real numbers, such that <math>xyz = 1</math> | |
+ | |||
+ | '''(i)''' Prove that; | ||
<math>\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1</math>. | <math>\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1</math>. | ||
− | |||
− | '''(ii)''' Prove that | + | '''(ii)''' Prove that <math>\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} = 1</math> for infinitely many triples of rational numbers <math>x</math>, <math>y</math> and <math>z</math>. |
[[2008 IMO Problems/Problem 2 | Solution]] | [[2008 IMO Problems/Problem 2 | Solution]] | ||
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[[2008 IMO Problems/Problem 3 | Solution]] | [[2008 IMO Problems/Problem 3 | Solution]] | ||
− | == Day | + | == Day 2 == |
=== Problem 4 === | === Problem 4 === | ||
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=== Problem 5 === | === Problem 5 === | ||
− | Let <math>n</math> and <math>k</math> be positive integers with <math>k \geq n</math> and <math>k - n</math> an even number. Let <math>2n</math> lamps labelled <math>1 | + | Let <math>n</math> and <math>k</math> be positive integers with <math>k \geq n</math> and <math>k - n</math> an even number. Let <math>2n</math> lamps labelled <math>1, 2, \dots, 2n</math> be given, each of which can be either ''on'' or ''off''. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). |
Let <math>N</math> be the number of such sequences consisting of <math>k</math> steps and resulting in the state where lamps <math>1</math> through <math>n</math> are all on, and lamps <math>n + 1</math> through <math>2n</math> are all off. | Let <math>N</math> be the number of such sequences consisting of <math>k</math> steps and resulting in the state where lamps <math>1</math> through <math>n</math> are all on, and lamps <math>n + 1</math> through <math>2n</math> are all off. | ||
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* [[2008 IMO]] | * [[2008 IMO]] | ||
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2008 IMO 2008 Problems on the Resources page] | * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2008 IMO 2008 Problems on the Resources page] | ||
+ | |||
+ | {{IMO box|year=2008|before=[[2007 IMO Problems]]|after=[[2009 IMO Problems]]}} |
Latest revision as of 23:30, 17 February 2021
Problems of the 49th IMO 2008 Spain.
Contents
[hide]Day 1
Problem 1
Let be the orthocenter of an acute-angled triangle . The circle centered at the midpoint of and passing through intersects line at points and . Similarly, define the points , , and .
Prove that six points , , , , and are concyclic.
Problem 2
Let be three real numbers, such that
(i) Prove that; .
(ii) Prove that for infinitely many triples of rational numbers , and .
Problem 3
Prove that there are infinitely many positive integers such that has a prime divisor greater than .
Day 2
Problem 4
Find all functions (so is a function from the positive real numbers) such that
for all positive real numbes satisfying
Problem 5
Let and be positive integers with and an even number. Let lamps labelled be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).
Let be the number of such sequences consisting of steps and resulting in the state where lamps through are all on, and lamps through are all off.
Let be number of such sequences consisting of steps, resulting in the state where lamps through are all on, and lamps through are all off, but where none of the lamps through is ever switched on.
Determine .
Problem 6
Let be a convex quadrilateral with different from . Denote the incircles of triangles and by and respectively. Suppose that there exists a circle tangent to ray beyond and to the ray beyond , which is also tangent to the lines and .
Prove that the common external tangents to and intersect on .
Resources
2008 IMO (Problems) • Resources | ||
Preceded by 2007 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2009 IMO Problems |
All IMO Problems and Solutions |