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Sxchen1110
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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 Apr 2024
0
4
1
+16
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
और पढ़ें ..
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Sxchen1110
8 Apr 2024
0
4
2
+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.
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Sxchen1110
8 Apr 2024
0
5
1
+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.
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Sxchen1110
31 Mar 2024