IMO 2008
 

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.[/quote]
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.[/quote]
3,
Prove that there are infinitely many positive integers n such that$ n^{2} + 1 $ has a prime divisor greater than$ 2n + \sqrt {2n}. $
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đề cũng không khó lắm .Không biết vòng 2 thế nào:hornytoro:

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IMO 08 ngày2
 

Q4: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 numbers w,x,y,z, satisfying wx = yz.

Q5:Let n and k be positive integers with k≥ 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 tgrough 2n are all off, but where none of the lamps n+1 through 2n is ever switched on.
Determine N/M

Q6:
Let ABCD be a convex quadrilateral with BA different from BC. Denote the incircles of triangles ABC and ADC by k1 and k2 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 k1 and k2 intersects on k.

Kết quả [Only registered and activated users can see links. Click Here To Register...] . Ba chú 42/42 : 2 China , 1 USA. Bé Thảo đoàn mình HCB , chúc mừng em.

3 ông 42/42 đều là Tàu cả ^:)^ , Trung Quốc vô đối ^:)^

[Only registered and activated users can see links. Click Here To Register...] . Năm ngoái chú Alex cũng Gold rồi.

Có 1 anh nhầm rồi:))

Where can I find original solutions (as in shortlist)?

You can find solution in IMO web,but shortlist ,you must wait,at least a few day before the next IMO

If you want to print the problems of IMO 2008 in high quality, you can download pdf-versions of text of problems (in english and vietnamese) here:

Có một lão gốc Trung Quốc nhưng thi cho đoàn Mỹ mà.:) (hình như thế)

Có ai làm dc bài 6 ko vậy,mình muốn xem lời giải của bài toán này quá!!!!!