HMMT 十一月 2017 · 冲刺赛 · 第 13 题

HMMT November 2017 — Guts Round — Problem 13

[ 5 ] 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 V cubic centimeters. Ritmo, on the other hand, measured each dimension to the nearest centimeter and multiplied the rounded measurements, getting a result of 2017 cubic centimeters. Find the positive difference between the least and greatest possible positive values for V . AB DA

解析

[ 5 ] 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 V cubic centimeters. Ritmo, on the other hand, measured each dimension to the nearest centimeter and multiplied the rounded measurements, getting a result of 2017 cubic centimeters. Find the positive difference between the least and greatest possible positive values for V . Proposed by: Yuan Yao Answer: 4035 The only possible way for Ritmo to get 2017 cubic centimeters is to have his measurements rounded to 1 , 1 , 2017 centimeters respectively. Therefore the largest value of V is achieved when the dimensions ′ ′ are (1 . 5 − )(1 . 5 − )(2017 . 5 − ) = 4539 . 375 − for some very small positive real , , and the smallest ′ value of V is achieved when the dimensions are (0 . 5 + )(0 . 5 + )(2016 . 5 + ) = 504 . 125 + for some ′ very small positive real , . Therefore the positive difference is 4539 − 504 = 4035. AB DA