MathDB

4

Part of ICMC 7

Problems(2)

Classical tetrahedron maximisation problem

Source: ICMC 7 Round 1 Problem 4

1/8/2024
Points A,B,C,A, B, C, and DD{} lie on the surface of a sphere with diameter 1. Determine the maximum possible volume of tetrahedron ABCD.ABCD.
Proposed by Fredy Yip
geometry3D geometryICMC
Convergence of infinite series

Source: ICMC 7 Round 2 Problem 4

3/12/2024
Let (tn)n⩾1(t_n)_{n\geqslant 1} be the sequence defined by t1=1,t2k=−tkt_1=1, t_{2k}=-t_k and t2k+1=tk+1t_{2k+1}=t_{k+1} for all k⩾1.k\geqslant 1. Consider the series ∑n=1∞tnn1/2024.\sum_{n=1}^\infty\frac{t_n}{n^{1/2024}}.Prove that this series converges to a positive real number.
Proposed by Dylan Toh
Convergenceseriesreal analysis