MathDB

7

Part of 2023 Harvard-MIT Mathematics Tournament

Problems(4)

Quadrilateral ABCD (2023 HMMT Geo #7)

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2/20/2023
Quadrilateral ABCDABCD is inscribed in circle Γ\Gamma. Segments ACAC and BDBD intersect at EE. Circle γ\gamma passes through EE and is tangent to Γ\Gamma at AA. Suppose the circumcircle of triangle BCEBCE is tangent to γ\gamma at EE and is tangent to line CDCD at CC. Suppose that Γ\Gamma has radius 33 and γ\gamma has radius 22. Compute BDBD.
geometrycyclic quadrilateral
HMMT Feb 2023 team p7

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2/20/2023
Let ABCABC be a triangle. Point DD lies on segment BCBC such that ∠BAD=∠DAC\angle BAD = \angle DAC. Point XX lies on the opposite side of line BCBC as AA and satisfies XB=XDXB=XD and ∠BXD=∠ACB\angle BXD = \angle ACB. The point YY is defined similarly. Prove that the lines XYXY and ADAD are perpendicular.
2023 Combinatorics #7

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2/28/2024
Svitlana writes the number 147147 on a blackboard. Then, at any point, if the number on the blackboard is nn, she can perform one of the following three operations: ∙\bullet if nn is even, she can replace nn with n2\frac{n}{2} ∙\bullet if nn is odd, she can replace nn with n+2552\frac{n+255}{2} and ∙\bullet if n≥64n \ge 64, she can replace nn with n−64n - 64. Compute the number of possible values that Svitlana can obtain by doing zero or more operations.
combinatorics
2023 Algebra NT #7 a+b = c+d

Source:

2/28/2024
If a,b,ca, b, c, and dd are pairwise distinct positive integers that satisfy lcm(a,b,c,d)<1000lcm (a, b, c, d) < 1000 and a+b=c+da+b = c+d, compute the largest possible value of a+ba + b.
number theory