HMMT 二月 2018 · 冲刺赛 · 第 5 题
HMMT February 2018 — Guts Round — Problem 5
题目详情
- [ 6 ] Randall proposes a new temperature system called Felsius temperature with the following conversion ◦ ◦ ◦ between Felsius E , Celsius C , and Fahrenheit F : ◦ ◦ 7 × C 7 × F − 80 ◦ E = + 16 = . 5 9 ◦ ◦ ◦ ◦ ◦ ◦ ◦ For example, 0 C = 16 E . Let x, y, z be real numbers such that x C = x E, y E = y F, z C = ◦ z F . Find x + y + z .
解析
- [ 6 ] Randall proposes a new temperature system called Felsius temperature with the following conversion ◦ ◦ ◦ between Felsius E , Celsius C , and Fahrenheit F : ◦ ◦ 7 × C 7 × F − 80 ◦ E = + 16 = . 5 9 ◦ ◦ ◦ ◦ ◦ ◦ ◦ For example, 0 C = 16 E . Let x, y, z be real numbers such that x C = x E, y E = y F, z C = ◦ z F . Find x + y + z . Proposed by: Yuan Yao Answer: − 120 ◦ ◦ ◦ Notice that (5 k ) C = (7 k + 16) E = (9 k + 32) F , so Felsius is an exact average of Celsius and Fahrenheit at the same temperature. Therefore we conclude that x = y = z , and it is not difficult to compute that they are all equal to − 40.