HMMT 十一月 2012 · 冲刺赛 · 第 33 题
HMMT November 2012 — Guts Round — Problem 33
题目详情
- [ 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 bouncesagainst walls exactly 2012 times before arriving at a vertex of the square.
HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND
∑
2012
英文原题
[ 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
解析
英文解析
- [ 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 meetsafter leaving the origin be ( a, b ), so that b > a . Note that a and b are coprime, otherwise the ball willreach a vertex before the 2012 th bounce. We wish to minimize the slope of the line to this point fromorigin, 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 isjust 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