HMMT 十一月 2015 · THM 赛 · 第 2 题
HMMT November 2015 — THM Round — Problem 2
题目详情
英文原题
- Consider a 2 × 2 grid of squares. David writes a positive integer in each of the squares. Next to eachrow, he writes the product of the numbers in the row, and next to each column, he writes the productof the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?
解析
英文解析
- Consider a 2 × 2 grid of squares. David writes a positive integer in each of the squares. Next to eachrow, he writes the product of the numbers in the row, and next to each column, he writes the productof the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?
Proposed by: Alexander Katz
Answer: 88
Let the four numbers be a, b, c, d , so that the other four numbers are ab, ad, bc, bd . The sum of theseeight numbers is a + b + c + d + ab + ad + bc + bd = ( a + c ) + ( b + d ) + ( a + c )( b + d ) = 2015, andso ( a + c + 1)( b + d + 1) = 2016. Since we seek to minimize a + b + c + d , we need to find the twofactors of 2016 that are closest to each other, which is easily calculated to be 42 · 48 = 2016; this makesa + b + c + d = 88 .