HMMT 十一月 2017 · 团队赛 · 第 5 题

HMMT November 2017 — Team Round — Problem 5

[ 35 ] Ashwin the frog is traveling on the xy -plane in a series of 2 − 1 steps, starting at the origin. th At the n step, if n is odd, then Ashwin jumps one unit to the right. If n is even, then Ashwin jumps m m units up, where m is the greatest integer such that 2 divides n . If Ashwin begins at the origin, 2016 what is the area of the polygon bounded by Ashwin’s path, the line x = 2 , and the x -axis?

解析

[ 35 ] Ashwin the frog is traveling on the xy -plane in a series of 2 − 1 steps, starting at the origin. th At the n step, if n is odd, then Ashwin jumps one unit to the right. If n is even, then Ashwin jumps m m units up, where m is the greatest integer such that 2 divides n . If Ashwin begins at the origin, 2016 what is the area of the polygon bounded by Ashwin’s path, the line x = 2 , and the x -axis? Proposed by: Nikhil Reddy 2015 2017 Answer: 2 · (2 − 2018) 2017 Notice that since v ( x ) = v (2 − x ), the path divides the rectangle bonded by the coordinate axes 2 2 and the two lines passing through Ashwin’s final location parallel to the axes. The answer is therefore half of the product of the coordinates of Ashwin’s final coordinates. The x -coordinate is the number of 2016 2017 odd number steps (which is 2 ). The y -coordinate is the number of total powers of 2 in (2 − 1)!. 2015 2017 The final answer is therefore 2 · (2 − 2018).