[問題] IMO 2008 day2

[boggart0803 (幻形怪)]

Problem 4

Find all functions f : (0,∞)→(0,∞) (so, f is a function from the
positive real numbers to the positive real numbers) such that

(f(w))^2+(f(x))^2     w^2+x^2
-----------------  =  -------
  f(y^2)+f(z^2)       y^2+z^2

for all positive real numbers w, x, y, z, satisfying wx=yz


Problem 5

Let n and k be positive integers with k>=n and k-n an even number.
Let 2n lamps labelled 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 the 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 of, but where none of the lamps n+1 through
2n is ever switched on.

Determine N/M


Problem 6

Let ABCD be a convex quadrilateral with BA different from BC. Denote
the incircles of triangles ABC and ADC by ω_1 and ω_2 respectively.
Suppose that there exists a circle ω 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 ω_1 and ω_2 intersect on ω.

真抱歉...擺了大烏龍@@
剛剛得到的題目來源是假的...

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推 myflame:有自信電子通訊設施防堵完善?XD
推 JGU:怎麼回事啊?
※ 編輯: boggart0803     來自: 220.133.169.67       (07/17 21:17)
※ 編輯: boggart0803     來自: 220.133.169.67       (07/17 21:36)
推 myflame:可以請問本來po的假4,5,6題出典為何?想寫看看^^
※ 編輯: boggart0803     來自: 220.133.169.67       (07/18 08:51)