PUMaC 2020 · 代数（B 组） · 第 2 题

PUMaC 2020 — Algebra (Division B) — Problem 2

Princeton has an endowment of 5 million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest a million dollars in the dorms, the students will spend an additional 5 a hours per week studying. If the university invests b million dollars in better food, the students will spend an additional 3 b hours per week studying. Finally, if 2 the c million dollars are invested in parties, students will be more relaxed and spend 11 c − c more hours per week studying. The university wants to invest its 5 million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study? p x + a

解析

Princeton has an endowment of 5 million dollars and wants to invest it into improving campus life. The university has three options: it can either invest in improving the dorms, campus parties or dining hall food quality. If they invest a million dollars in the dorms, the students will spend an additional 5 a hours per week studying. If the university invests b million dollars in better food, the students will spend an additional 3 b hours per week studying. Finally, if 2 the c million dollars are invested in parties, students will be more relaxed and spend 11 c − c more hours per week studying. The university wants to invest its 5 million dollars so that the students get as many additional hours of studying as possible. What is the maximal amount that students get to study? Proposed by: Aleksa Milojevic Answer: 34 Note: On the original algebra test, we had forgotten the “per week” on the last quantity of hours. 2 We have to maximize 5 a + 3 b + 11 c − c when a + b + c = 5. It is clear that b = 0 as it 2 2 is always better to invest in a than in b . We know 5 a + 11 c − c = 5( a + c ) + (6 c − c ) = 2 2 5 · 5 + 9 − ( c − 3) = 34 − ( c − 3) . The optimal choice is a = 2 , b = 0 , c = 3; our answer is 34 . p x + a