HMMT 十一月 2017 · 冲刺赛 · 第 3 题

HMMT November 2017 — Guts Round — Problem 3

[ 2 ] The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to k > 0, find k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2017, November 11, 2017 — GUTS ROUND Organization Team Team ID#

解析

[ 2 ] The length of a rectangle is three times its width. Given that its perimeter and area are both numerically equal to k > 0, find k . Proposed by: Yuan Yao 64 Answer: 3 Let a be the width of the rectangle. Then the length of the rectangle is 3 a , so the perimeter is 2 2( a + 3 a ) = 8 a , and the area is 3 a . Since the length is numerically equal to the width, we know that 2 8 a = 3 a = k. 64 8 Because k > 0, the rectangle is non-degenerate. It follows that 8 = 3 a , so a = . Therefore, k = . 3 3