作者FAlin (FA（バルシェ应援）)
看板IMO_Taiwan
标题[问题] IMO 2012 in Argentina Day 2
时间Thu Jul 12 03:26:49 2012
4. Find all functions f: Z→Z, such that for all a+b+c = 0 holds :
f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a)
5. Let △ABC be a triangle with ∠C = 90度 and that D be the foot of the
altitude from C. Let X be a point in the interior of the segment CD.
Let K be the point on the segment AX such that BK = BC. Similarly,
let L be the point on the segment BX such that AL = AC.
Let M be the point of intersection of AL and BK.
Show that MK = ML.
6. Determine all positive integers n for which there exist non-negative
integers a_1, a_2, ...,a_n such that:
1 1 1 1 2 n
----- + ----- + ... + ----- = ----- + ----- + ... + ----- = 1
2^a_1 2^a_2 2^a_n 3^a_1 3^a_2 3^a_n
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※ 编辑: FAlin 来自: 140.112.244.138 (07/12 03:28)
1F：推 darkseer:有没有人可以开示第五题难不难XDDD 对我来说实在太难了 07/12 08:16
2F：推 Dawsen:第五题可以用暴力法 07/12 12:33
3F：推 hahaj6u4503:解三个点XD 07/12 12:52
4F：推 hahaj6u4503:第六题没有想像中难, 但就差个临门一脚的构造 > < 07/12 12:57
5F：→ FAlin:第五题很美 07/12 13:38
6F：推 hahaj6u4503:画了一堆圈圈与辅助线图整个超漂亮!!!!! 07/12 16:54