MathDB

7

Part of 2019 Harvard-MIT Mathematics Tournament

Problems(4)

HMMT Team 2019/7: Fat polygons are big

Source:

2/17/2019
A convex polygon on the plane is called wide if the projection of the polygon onto any line in the same plane is a segment with length at least 1. Prove that a circle of radius 13\tfrac{1}{3} can be placed completely inside any wide polygon.
HMMTcombinatoricsgeometry
HMMT Algebra/NT 2019/7: A non-symmetric three-variable infinite sum

Source:

2/17/2019
Find the value of ∑a=1∞∑b=1∞∑c=1∞ab(3a+c)4a+b+c(a+b)(b+c)(c+a).\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.
HMMTalgebraSummation
HMMT Combinatorics 2019/7: Putnam 2004 A1 for grownups

Source:

2/17/2019
In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a 23\tfrac{2}{3} probability of voting for Alice and a 13\tfrac{1}{3} probability of voting for Celia.
What is the probability that Alice wins the election (by having more votes than Celia)?
HMMTcombinatoricsalgebra
HMMT Geometry 2019/7: Do you remember your triangle geometry?

Source:

2/17/2019
Let ABCABC be a triangle with AB=13AB = 13, BC=14BC = 14, CA=15CA = 15. Let HH be the orthocenter of ABCABC. Find the radius of the circle with nonzero radius tangent to the circumcircles of AHBAHB, BHCBHC, CHACHA.
HMMTgeometry