MathDB

3

Part of 2022 Harvard-MIT Mathematics Tournament

Problems(4)

2022 Team 3

Source:

3/14/2022
Let triangle ABCABC be an acute triangle with circumcircle Γ\Gamma. Let XX and YY be the midpoints of minor arcs ABAB and ACAC of Γ\Gamma, respectively. If line XYXY is tangent to the incircle of triangle ABCABC and the radius of Γ\Gamma is RR, find, with proof, the value of XYXY in terms of RR.
geometry
2022 Algebra/NT #3

Source:

3/11/2022
Let x1,x2,...,x2022x_1, x_2, . . . , x_{2022} be nonzero real numbers. Suppose that xk+1xk+1<0x_k + \frac{1}{x_{k+1}} < 0 for each 1≤k≤20221 \leq k \leq 2022, where x2023=x1x_{2023}=x_1. Compute the maximum possible number of integers 1≤n≤20221 \leq n \leq 2022 such that xn>0x_n > 0.
algebra
2022 Geometry 3

Source:

3/14/2022
Let ABCDABCD and AEFGAEF G be unit squares such that the area of their intersection is 2021\frac{20}{21} . Given that ∠BAE<45o\angle BAE < 45^o, tan⁡∠BAE\tan \angle BAE can be expressed as ab\frac{a}{b} for relatively prime positive integers aa and bb. Compute 100a+b100a + b.
geometry
2022 Combinatorics 3

Source:

3/18/2022
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly 1010 operations.
combinatorics