MathDB

1

Part of 2017 239 Open Mathematical Olympiad

Problems(2)

orthocentre gives an altitude

Source: 239 2017 S1

5/22/2020
On the side ACAC of triangle ABCABC point DD is chosen. Let I1,I2,II_1, I_2, I be the incenters of triangles ABD,BCD,ABCABD, BCD, ABC respectively. It turned out that II is the orthocentre of triangle I1I2BI_1I_2B. Prove that BDBD is an altitude of triangle ABCABC.
geometry
An Average on Permutations

Source: 239 2017 J1

5/31/2020
Denote every permutation of 1,2,…,n1,2,\dots, n as σ=(a1,a2,…,n)\sigma =(a_1,a_2,\dots,n). Prove that the sum ∑1(a1)(a1+a2)(a1+a2+a3)…(a1+a2+⋯+an)\sum \frac{1}{(a_1)(a_1+a_2)(a_1+a_2+a_3)\dots(a_1+a_2+\dots+a_n)} taken over all possible permutations σ\sigma equals 1n!\frac{1}{n!}.
combinatoricspermutations