MathDB

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Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

Hexagon Areas (2023 Geo #1 HMMT Feb)

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2/23/2023
Let ABCDEFABCDEF be a regular hexagon, and let PP be a point inside quadrilateral ABCDABCD. If the area of triangle PBCPBC is 2020, and the area of triangle PADPAD is 2323, compute the area of hexagon ABCDEFABCDEF.
geometryHMMT
HMMT Feb 2023 Team p1

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2/20/2023
For any positive integer aa, let τ(a)\tau(a) be the number of positive divisors of aa. Find, with proof, the largest possible value of 4τ(n)−n4\tau(n)-n over all positive integers nn.
2023 Combinatorics #1

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4/17/2023
There are 800800 marbles in a bag. Each marble is colored with one of 100100 colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.
Suppose Anna has not stopped after drawing 699699 marbles. Compute the probability that she stops immediately after drawing the 700700th marble.
2023 Algebra/NT #1

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4/17/2023
Suppose P(x)P(x) is a cubic polynomial with integer coefficients such P(5)=5P(\sqrt{5})=5 and P(53)=553P(\sqrt[3]{5})=5\sqrt[3]{5}.