HMMT 十一月 2016 · GEN 赛 · 第 3 题

HMMT November 2016 — GEN Round — Problem 3

Let V be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area x , where x is not equal to 48 or 240. What is the sum of all possible values of x ?

解析

Let V be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area x , where x is not equal to 48 or 240. What is the sum of all possible values of x ? Proposed by: Eshaan Nichani Answer: 260 Let the length, width, and height of the prism be s , s , s . WIthout loss of generality, assume that 1 2 3 s ≤ s ≤ s . Then, we have that s s = 48 and s s = 240. Noting that s ≤ s , we must have 1 2 3 1 2 2 3 1 2 ( s , s ) = (1 , 48) , (2 , 24) , (3 , 16) , (4 , 12) , (6 , 8). We must also have s s = 240 and s ≤ s , and the 1 2 2 3 2 3 only possibilities for ( s , s ) that yield integral s that satisfy these conditions are (4 , 12), which gives 1 2 3 s = 20, and (6 , 8), which gives s = 30. Thus, the only valid ( s , s , s ) are (4 , 12 , 20) and (6 , 8 , 30). 3 3 1 2 3 It follows that the only possible areas of the third face are 4(20) = 80 and 6(30) = 180, so the desired answer is 80 + 180 = 260 .