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Difference between revisions of "2008 IMO Problems"

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[[
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Problems of the 49th [[IMO]] 2008 Spain.
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== Link title ==
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<math>[Insert formula here]<math><nowiki>Insert formula here</nowiki>
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== Day I ==
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----
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=== Problem 1 ===
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== [[Image:Headline text]]<nowiki><dollar/>Insert non-formatted text here<dollar/>[[Image:[Example.jpg]]]</nowiki> ==
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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 the sideline <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>.
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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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[[2008 IMO Problems/Problem 1 | Solution]]
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=== Problem 2 ===
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'''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real numbers, all different from <math>1</math>, such that <math>xyz = 1</math>, then prove that
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<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>.
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(With the <math>\sum</math> sign for cyclic summation, this inequality could be rewritten as <math>\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1</math>.)
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'''(ii)''' Prove that equality is achieved for infinitely many triples of rational numbers <math>x</math>, <math>y</math> and <math>z</math>.
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[[2008 IMO Problems/Problem 2 | Solution]]
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=== Problem 3 ===
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Prove that there are infinitely many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater than <math>2n + \sqrt {2n}</math>.
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[[2008 IMO Problems/Problem 3 | Solution]]
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== Day II ==
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=== Problem 4 ===
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Find all functions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that
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<center>
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<math>\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}</math>
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</center>
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for all positive real numbes <math>w,x,y,z,</math> satisfying <math>wx = yz.</math>
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[[2008 IMO Problems/Problem 4 | Solution]]
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=== Problem 5 ===
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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</math>, <math>2</math>, ..., <math>2n</math> be given, each of which can be either [i]on[/i] or [i]off[/i]. 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).
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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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Let <math>M</math> be number of such sequences consisting of <math>k</math> steps, 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, but where none of the lamps <math>n + 1</math> through <math>2n</math> is ever switched on.
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Determine <math>\frac {N}{M}</math>.
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[[2008 IMO Problems/Problem 5 | Solution]]
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=== Problem 6 ===
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Let <math>ABCD</math> be a convex quadrilateral with <math>BA</math> different from <math>BC</math>. Denote the incircles of triangles <math>ABC</math> and <math>ADC</math> by <math>k_{1}</math> and <math>k_{2}</math> respectively. Suppose that there exists a circle <math>k</math> tangent to ray <math>BA</math> beyond <math>A</math> and to the ray <math>BC</math> beyond <math>C</math>, which is also tangent to the lines <math>AD</math> and <math>CD</math>.  
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Prove that the common external tangents to <math>k_{1}</math> and <math>k_{2}</math> intersects on <math>k</math>.
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[[2008 IMO Problems/Problem 6 | Solution]]
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== Resources ==
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* [[2008 IMO]]
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* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2008 IMO 1981 Problems on the Resources page]

Revision as of 10:28, 5 August 2008

Problems of the 49th IMO 2008 Spain.

Day I

Problem 1

Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$. Similarly, define the points $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$.

Prove that six points $A_{1}$ , $A_{2}$, $B_{1}$, $B_{2}$, $C_{1}$ and $C_{2}$ are concyclic.

Solution

Problem 2

(i) If $x$, $y$ and $z$ are three real numbers, all different from $1$, such that $xyz = 1$, then prove that $\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$. (With the $\sum$ sign for cyclic summation, this inequality could be rewritten as $\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1$.)

(ii) Prove that equality is achieved for infinitely many triples of rational numbers $x$, $y$ and $z$.

Solution

Problem 3

Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$.

Solution

Day II

Problem 4

Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that

$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}$

for all positive real numbes $w,x,y,z,$ satisfying $wx = yz.$

Solution

Problem 5

Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1$, $2$, ..., $2n$ be given, each of which can be either [i]on[/i] or [i]off[/i]. 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 $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off.

Let $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on.

Determine $\frac {N}{M}$.

Solution

Problem 6

Let $ABCD$ be a convex quadrilateral with $BA$ different from $BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $k_{1}$ and $k_{2}$ respectively. Suppose that there exists a circle $k$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$.

Prove that the common external tangents to $k_{1}$ and $k_{2}$ intersects on $k$.

Solution

Resources

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