[問題] 58th IMO in Rio Day 2
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.
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※ 編輯: yclinpa (112.105.246.4), 07/20/2017 06:49:40
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