[问题] 58th IMO in Rio Day 2

楼主: yclinpa (一等士官长 薇楷的爹)   2017-07-20 06:48:48
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.
作者: Dawsen (好友名单不见了啦...)   2017-07-22 12:53:00
看统计资料 今年应该是史上最难一年吧
作者: darkseer   2017-07-23 03:22:00
确实,现在我看题目都分不出来了我会做P3, P6,但不会做P2 XD
作者: cmrafsts (喵喵)   2017-07-23 08:20:00
P2也只能 用力做不如试试P5?
作者: Dawsen (好友名单不见了啦...)   2017-07-23 11:40:00
darkseer 会做 P3, P6 不意外... 但不会做 P2 颇令人意外

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