HMMT 十一月 2023 · GEN 赛 · 第 4 题

HMMT November 2023 — GEN Round — Problem 4

Suppose that a and b are real numbers such that the line y = ax + b intersects the graph of y = x at two distinct points A and B . If the coordinates of the midpoint of AB are (5 , 101) , compute a + b .

解析

Suppose that a and b are real numbers such that the line y = ax + b intersects the graph of y = x at two distinct points A and B . If the coordinates of the midpoint of AB are (5 , 101) , compute a + b . Proposed by: Rishabh Das Answer: 61 2 2 2 Solution 1: Let A = ( r, r ) and B = ( s, s ) . Since r and s are roots of x − ax − b with midpoint 5 , r + s = 10 = a (where the last equality follows by Vieta’s formula). Now, as − rs = b (Vieta’s formula), observe that 2 2 2 202 = r + s = ( r + s ) − 2 rs = 100 + 2 b. This means b = 51 , so the answer is 10 + 51 = 61 . 2 2 Solution 2: As in the previous solution, let A = ( r, r ) and B = ( s, s ) and note r + s = 10 = a . Fixing a = 10 , the y -coordinate of the midpoint is 50 when b = 0 (and changing b shifts the line up or down by its value). So, increasing b by 51 will make the midpoint have y -coordinate 50 + 51 = 101 , so the answer is 10 + 51 = 61 .