MathDB

6

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

hidden triple angle???(2023 HMMT G6)

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2/21/2023
Convex quadrilateral ABCDABCD satisfies ∠CAB=∠ADB=30∘,∠ABD=77∘,BC=CD\angle{CAB} = \angle{ADB} = 30^{\circ}, \angle{ABD} = 77^{\circ}, BC = CD and ∠BCD=n∘\angle{BCD} =n^{\circ} for some positive integer nn. Compute nn.
HMMT Feb 2023 team p6

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2/20/2023
For any odd positive integer nn, let r(n)r(n) be the odd positive integer such that the binary representation of r(n)r(n) is the binary representation of nn written backwards. For example, r(2023)=r(111111001112)=111001111112=1855r(2023)=r(111111001112)=111001111112=1855. Determine, with proof, whether there exists a strictly increasing eight-term arithmetic progression a1,…,a8a_1, \ldots, a_8 of odd positive integers such that r(a1),…,r(a8)r(a_1), \ldots , r(a_8) is an arithmetic progression in that order.
grid P(6) (2023 HMMT combi )

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2/28/2023
Each cell of a 33 × 33 grid is labeled with a digit in the set {1,2,3,4,51, 2, 3, 4, 5} Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 11 to 55 is recorded at least once.
gridHMMT
2023 Algebra NT #6 a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)}

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2/28/2024
Suppose a1,a2,...,a100a_1, a_2, ... , a_{100} are positive real numbers such that ak=kak−1ak−1−(k−1)a_k =\frac{ka_{k-1}}{a_{k-1} - (k - 1)} for k=2,3,...,100k = 2, 3, ... , 100. Given that a20=a23a_{20} = a_{23}, compute a100a_{100}.
algebra