MathDB

6

Part of 2022 Stanford Mathematics Tournament

Problems(4)

SMT 2022 Algebra #6

Source:

3/27/2023
Compute cot⁡(∑n=123cot⁡−1(1+∑k=1n2k)).\cot\left(\sum_{n=1}^{23}\cot^{-1}\left(1+\sum_{k=1}^n2k\right)\right).
SMT 2022 Calculus #6

Source:

3/29/2023
Let f(x)=cos⁡(x3−4x2+5x−2).f(x)=\cos(x^3-4x^2+5x-2). If we let f(k)f^{(k)} denote the kkth derivative of ff, compute f(10)(1)f^{(10)}(1). For the sake of this problem, note that 10!=362880010!=3628800.
SMT 2022 Discrete #6

Source:

4/1/2023
Let A\mathcal{A} be the set of finite sequences of positive integers a1,a2,…,aka_1,a_2,\dots,a_k such that ∣an−an−1∣=an−2|a_n-a_{n-1}|=a_{n-2} for all 3⩽n⩽k3\leqslant n\leqslant k. If a1=a2=1a_1=a_2=1, and k=18k=18, determine the number of elements of A\mathcal{A}.
SMT 2022 Geometry #6

Source:

4/1/2023
Let the incircle of △ABC\triangle ABC be tangent to AB,BC,ACAB,BC,AC at points M,N,PM,N,P, respectively. If ∡BAC=30∘\measuredangle BAC=30^\circ, find [BPC][ABC]⋅[BMC][ABC]\tfrac{[BPC]}{[ABC]}\cdot\tfrac{[BMC]}{[ABC]}, where [ABC][ABC] denotes the area of △ABC\triangle ABC.