HMMT 十一月 2015 · THM 赛 · 第 2 题

HMMT November 2015 — THM Round — Problem 2

Consider a 2 × 2 grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of 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

Consider a 2 × 2 grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of 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 these eight numbers is a + b + c + d + ab + ad + bc + bd = ( a + c ) + ( b + d ) + ( a + c )( b + d ) = 2015, and so ( a + c + 1)( b + d + 1) = 2016. Since we seek to minimize a + b + c + d , we need to find the two factors of 2016 that are closest to each other, which is easily calculated to be 42 · 48 = 2016; this makes a + b + c + d = 88 .