PUMaC 2019 · 团队赛 · 第 8 题

PUMaC 2019 — Team Round — Problem 8

The curves y = x + 5 and y = x − 3 x intersect at points A and B . C is a point on the lower curve between A and B . The maximum possible area of the quadrilateral ABCO can A be written as for coprime A, B . Find A + B . B √ 5 5 5 5 5

解析

The curves y = x + 5 and y = x − 3 x intersect at points A and B . C is a point on the lower curve between A and B . The maximum possible area of the quadrilateral ABCO can A be written as for coprime A, B . Find A + B . B Proposed by: Yuxi Zheng Answer: 253 2 y = x + 5 and y = x − 3 x intersect at points A ( − 1 , 4) and B (5 , 10). Segment OB is a straight 2 line through the origin with the equation y = 2 x . Let C ( x, x − 3 x ) be a point on the curve 2 between O and B . The vertical distance between point C and segment OB is 2 x − ( x − 3 x ). 1 2 The area of triangle OBC can be obtained by × ( x − x ) × (2 x − ( x − 3 x )), where x B O B 2 and x denote the x coordinates of point B and point O respectively. Therefore, the area of O 1 2 triangle OBC equals × 5 × (2 x − ( x − 3 x )). After completing the squre we can get that 2 5 5 125 the maximum area of OBC happens when C is ( , − ), and the area is . The intersection 2 4 8 of AB with the y axis is (0 , 5), therefore the area of triangle OAB can be easily calculated as 1 245 × ( x − x ) × 5 = 15 . Therefore, the area of OABC is simply . B A 2 8 √ 5 5 5 5 5