4. Let R and S be different points on a circle Ω such that RS is not a diameter. Let l be the tangent line to Ω at R. Point T is such that S is the midpoint of the line segment RT. Point J is chosen on the shorter arc RS of Ω so that the circumcircle Γ of triangle JST intersects l at two distinct points. Let A be the common point of Γ and l that is closer to R. Line AJ meets Ω again at K. Prove that the line KT is tangent to Γ. 5. An integer N ≧ 2 is given. A collection of N(N+1) soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove N(N-1) players from this row leaving a new row of 2N players in which the following N conditions hold: (1) no one stands between the two tallest players, (2) no one stands between the third and the fourth tallest players, ... (N) no one stands between the two shortest players. Show that this is always possible. 6. An ordered pair (x,y) of integers is a primitive point if the greatest common divisor of x and y is 1. Given a finite set S of primitive points, prove that there exists a positive integer n and integers a_0, a_1, ..., a_n such that, for each (x,y) in S, we have: a_0 x^n + a_1 x^{n-1} y + a_2 x^{n-2} y^2 + ... + a_{n-1} x y^{n-1} + a_n y^n = 1. -- 雄兔腳撲朔 雌兔眼迷離 兩兔傍地走 安能辨我是雄雌 --

※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 112.105.246.4 ※ 文章網址: https://webptt.com/m.aspx?n=bbs/IMO_Taiwan/M.1500504535.A.617.html ※ 編輯: yclinpa (112.105.246.4), 07/20/2017 06:49:40