HMMT 十一月 2016 · 冲刺赛 · 第 20 题

英文解析

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.
    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
    √ √ √
    √
    ( )
    2016 120 120 7500 7500 65
    5 5 5 < 2016 ⇒ < < = = < .5
    4 15
    5 2015 · 2014 · 2013 · 2012 2000 10 1000 1000
    Therefore, the volume of R is lower-bounded by
    ( )
    11 88
    2 · 2 45 − = 360 − > 3552
    89 89
    and upper-bounded by
    ( ) ( )
    2 2
    0 . 8 6
    2 1 + · 2(45) = 360 1 + < 365 .2
    100 1000
    Thus, R rounded to the nearest ten is 360.