HMMT 十一月 2016 · 冲刺赛 · 第 20 题

HMMT November 2016 — Guts Round — Problem 20

[ 11 ] Let V be the volume enclosed by the graph 2016 2016 2 x + y + z = 2016 Find V rounded to the nearest multiple of ten.

解析

[ 11 ] Let V be the volume enclosed by the graph 2016 2016 2 x + y + z = 2016 Find V rounded to the nearest multiple of ten. Proposed by: Henrik Boecken Answer: 360 Let R be the region in question. Then we have √ √ √ √ √ √ 2016 2016 2 2 [ − 1 , 1] × [ − 2014 , 2014] ⊂ R ⊂ [ − 2016 , 2016] × [ 2016 , 2016] We find some bounds: we have √ √ 2016 < 2025 = 45 . √ By concavity of · , we have the bound √ √ √ 11 78 11 2014 ≤ 1936 + 2025 = 45 − . 89 89 89 √ 2016 2016 Finally, if we let 2016 = 1 + , then (1 + ) = 2016, so √ √ √ √ ( ) 5 2016 120 120 7500 7500 6 5 5 5 5 < 2016 ⇒ < < = = < . 4 15 5 2015 · 2014 · 2013 · 2012 2000 10 1000 1000 Therefore, the volume of R is lower-bounded by ( ) 11 88 2 2 · 2 45 − = 360 − > 355 89 89 and upper-bounded by ( ) ( ) 2 2 0 . 8 6 2 2 1 + · 2(45) = 360 1 + < 365 . 100 1000 Thus, R rounded to the nearest ten is 360.