เครดิตข้อสอบจาก
mathlinks
ข้อสอบวันแรก 16 กรกฎาคม 2008
IMO 2008, Question 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.
Author: Andrey Gavrilyuk, Russia
IMO 2008, Question 2
(i) If $x,\ y$ and $z$ are real numbers, different from 1, such that $xyz = 1$ prove that $$\sum \frac {x^{2}}{(x - 1)^{2}} \geq 1$$
(ii) Prove that equality case is achieved for infinitely many triples of rational numbers $x,\ y$ and $z$.
Author: Walther Janous, Austria
IMO 2008, Question 3
Prove that there are infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{2n}$.
Author: Kęstutis Česnavičius, Lithuania
ข้อสอบวันที่สอง 17 กรกฎาคม 2008
IMO 2008, Question 4
Find all functions $f: (0, \infty) \mapsto (0, \infty)$ (so $f$ is a function from the positive real numbers) such that
$$\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{\left( f(y) \right)^2 + \left( f(z) \right)^2} = \frac {w^2 + x^2}{y^2 + z^2}$$
for all positive real numbers $w,x,y,z,$ satisfying $wx = yz$.
Author: Hojoo Lee, South Korea
IMO 2008, Question 5
Let $n$ and $k$ be positive integers with $k\ge n$ and $k-n$ an even number. Let $2n$ lamps labeled $1, 2, ...,2n$ 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 $N$ 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 $N/M$.
Author: Bruno Le Floch and Ilia Smilga, France
IMO 2008, Question 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 $k$.
Author: Vladimir Shmarov, Russia