MathDB

2

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

2023 Geometry #2

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4/12/2023
Points XX, YY, and ZZ lie on a circle with center OO such that XY=12XY=12. Points AA and BB lie on segment XYXY such that OA=AZ=ZB=BO=5OA=AZ=ZB=BO=5. Compute ABAB.
HMMT Feb 2023 Team p2

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2/20/2023
Prove that there do not exist pairwise distinct complex numbers a,b,c,a, b, c, and dd such that a3−bcd=b3−acd=c3−abd=d3−abc.a^3-bcd=b^3-acd=c^3-abd=d^3-abc.
2023 Combinatorics #2

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4/17/2023
Compute the number of ways to tile a 3×53\times5 rectangle with one 1×11\times1 tile, one 1×21\times2 tile, one 1×31\times3 tile, one 1×41\times4, and one 1×51\times5 tile. (The tiles can be rotated, and tilings that differ by rotation or reflection are considered distinct.)
2023 Algebra/NT #2

Source:

4/17/2023
Compute the number of positive integers n≤1000n\le1000 such that lcm(n,9)\text{lcm}(n,9) is a perfect square. (Recall that lcm\text{lcm} denotes the least common multiple.)