作者FAlin (FA（バルシェ应援）)
看板IMO_Taiwan
标题[问题] IMO 2013 in Colombia Day1
时间Wed Jul 24 09:42:01 2013
1. Prove that for any two positive integers k , n there exist positive
integers m_1 , m_2 , ... , m_k such that
2^k - 1 1 1 1
1 + ------- = ( 1 + --- )( 1 + --- )...( 1 + --- ) .
n m_1 m_2 m_k
2. Giver 2013 red and 2014 blue points in the plane , no three of them on a
line. We aim to split plane by lines (not passing through these points)
into regions such that there are no regions containing points of both the
colors. What is the least number of lines that always suffice?
3. Let ABC be a triangle and that A_1 , B_1 , and C_1 be points of cantact of
the excircles with the sides BC , AC , and AB , respectively. Prove that if
the circumcenter of △A_1B_1C_1 lies on the circumcircle of △ABC , then
△ABC is a right triangle.
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◆ From: 124.11.128.7
1F：推 Dawsen:有请各位学长做难度分析~ 07/24 22:00
2F：→ FAlin:第三题没难度! 07/24 23:13
3F：推 cmrafsts:今年TWN1可能BRONZE 07/25 04:35
4F：推 cmrafsts:第三题用geogebra是没难度，但现场做会画不准... 07/25 09:58
5F：推 myflame:1还算简单 写起来有点费事就是(或者我的方法不够好) 07/25 23:29
6F：推 Dawsen:我想知道楼上的解法，写起来费事的应该是妙解 07/26 05:29
7F：推 myflame:ok 这几天没电脑 晚点补上 07/27 13:22
8F：推 cuttlefish:原po不是冥灯大大吗?xd 08/01 03:06