※ 引述《FAlin (FA（バルシェ應援）)》之銘言： : 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. --------------------------------------防第一題雷--------------------------- k=1取m1=n k>1取m1=n，如果n是奇數 m1=n+2^k-2，如果n是偶數 除掉後就可以化為k比較小的狀況，by induction and we are done. --------------------------------------------------------------------------- 這題另一個想法是希望那k個分數可以以某種方法通分，使恰有一個分子比分母多2^i i=1,2,...,k-1 而且可以對消。像k=3 n=4l+1時可以用 4l+2 4l+4 4l+8 ------ ------ ------ 4l+1 4l+2 4l+4 所以大概可以直接構造m1~mk。 台灣隊兩種方法各有一半人用。 --------------------------------底部防雷---------------------------------- --

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