Sxchen1110

4 Questions
0 Answers

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+16

Algebra

For every integer n≥1, let P(n) denote the product of the first n odd numbers:
P(n)=1×3×5×⋯×(2n−1)
Prove that for all integers n≥1, P(n)=(2n)!/2^n×n!.

Sxchen1110 18 abr 2024

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Combinatorics

Let S = {1, 2, . . . , 2021}, and let F denote the set of functions f : S → S. For a function f ∈ F, let Tf = f 2021(s) : s ∈ S , where f 2021(s) denotes f(f(· · ·(f(s))· · ·)) with 2021 copies of f. Compute the remainder when sigma notation f∈F lee mas ..

Sxchen1110 8 abr 2024

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+16

Geometry

Let ABTCD be a convex pentagon with area 22 such that AB = CD and the circumcircles of triangles TAB and TCD are internally tangent. Given that ∠ATD = 90◦ , ∠BTC = 120◦ , BT = 4, and CT = 5, compute the area of triangle TAD.

Sxchen1110 8 abr 2024

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3

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+16

Geometry

Let the incircle of △ABC be tangent to AB, BC, AC at points M, N, P, respectively. If ∡BAC = 30◦ , find [BP C] [ABC] · [BMC] [ABC] , where [ABC] denotes the area of △ABC.

Sxchen1110 31 mar 2024