HMMT 十一月 2012 · 冲刺赛 · 第 33 题

HMMT November 2012 — Guts Round — Problem 33

[ 17 ] You are playing pool on a 1 × 1 table, and the cue ball is at the bottom left corner of the square. ◦ Find the smallest angle larger than 45 at which you could shoot the ball such that the ball bounces against walls exactly 2012 times before arriving at a vertex of the square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND ∑ 2012

解析

[ 17 ] − 1 1009 Answer: tan ( ) As per usual with reflection problems instead of bouncing off the sides of 1005 a 1 × 1 square we imagine the ball to travel in a straight line from origin in an infinite grid of 1 × 1 squares, ”bouncing” every time it meets a line x = m or y = n . Let the lattice point it first meets after leaving the origin be ( a, b ), so that b > a . Note that a and b are coprime, otherwise the ball will reach a vertex before the 2012th bounce. We wish to minimize the slope of the line to this point from origin, which is b/a . Now, the number of bounces up to this point is a − 1 + b − 1 = a + b − 2, so the given statement is just a + b = 2014. To minimize b/a with a and b relatively prime, we must have a = 1005, b = 1009, 1009 − 1 so that the angle is tan ( ). 1005