HMMT 十一月 2017 · 冲刺赛 · 第 25 题

HMMT November 2017 — Guts Round — Problem 25

[ 15 ] Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of 2017 cubic centimeters. Ritmo, on the other hand, measured each dimension to the nearest centimeter and multiplied the rounded measurements, getting a result of V cubic centimeters. Find the positive difference between the least and greatest possible positive values for V . AB DA

解析

[ 15 ] Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with infinite precision, multiplied the results and rounded the product to the nearest cubic centimeter, getting a result of 2017 cubic centimeters. Ritmo, on the other hand, measured each dimension to the nearest centimeter and multiplied the rounded measurements, getting a result of V cubic centimeters. Find the positive difference between the least and greatest possible positive values for V . Proposed by: Yuan Yao Answer: 7174 It is not difficult to see that the maximum possible value of V can be achieved when the dimensions ′ ′′ ′ ′′ are (0 . 5 + ) × (0 . 5 + ) × (8070 − ) = 2017 . 5 − for some very small reals , , > 0, which when measured by Ritmo, gives V = 1 · 1 · 8070 = 8070. Similarly, the minimum possible positive value of 8066 ′ ′′ V can be achieved when the dimensions are (1 . 5 − ) × (1 . 5 − ) × ( + ) = 2016 . 5 + for some 9 ′ ′′ very small reals , , > 0, which when measured by Ritmo, gives V = 1 · 1 · 896 = 896. Therefore, the difference between the maximum and minimum is 8070 − 896 = 7174. AB DA