MathDB

B2

Part of 2020 Princeton University Math Competition

Problems(4)

2020 PUMaC Algebra B2

Source:

1/1/2022
Princeton has an endowment of 55 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 aa million dollars in the dorms, the students will spend an additional 5a5a hours per week studying. If the university invests bb million dollars in better food, the students will spend an additional 3b3b hours per week studying. Finally, if the cc million dollars are invested in parties, students will be more relaxed and spend 11c−c211c - c^2 more hours per week studying. The university wants to invest its 55 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?
algebra
2020 PUMaC Geometry B2

Source:

12/31/2021
Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least 66 feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least 66 feet apart from each other?
geometry
2020 PUMaC NT B2

Source:

1/1/2022
Last year, the U.S. House of Representatives passed a bill which would make Washington, D.C. into the 5151st state. Naturally, the mathematicians are upset that Congress won’t prioritize mathematical interest of flag design in choosing how many U.S. states there should be. Suppose the U.S. flag must contain, as it does now, stars arranged in rows alternating between nn and n−1n - 1 stars, starting and ending with rows of n stars, where n≥2n \ge 2 is some integer and the flag has more than one row. What is the minimum number of states that the U.S. would need to contain so that there are at least three different ways, excluding rotations, to arrange the stars on the flag?
number theory
2020 PUMaC Individual Finals B2

Source:

1/1/2022
Prove that there is a positive integer MM for which the following statement holds: For all prime numbers pp, there is an integer nn for which p≤n≤Mp\sqrt{p} \le n \le M\sqrt{p} and pmod  n≤n2020p \mod n \le \frac{n}{2020} .
Note: Here, pmod  np \mod n denotes the unique integer r∈0,1,...,n−1r \in {0, 1, ..., n - 1} for which n∣p−rn|p -r. In other words, pmod  np \mod n is the residue of pp upon division by nn.
number theory