MathDB

2

Part of 2021 South East Mathematical Olympiad

Problems(2)

Centers in a isosceles triangle

Source: 2021China South East Mathematical Olympiad Grade10 P2

7/28/2021
In △ABC\triangle ABC,AB=AC>BCAB=AC>BC, point O,HO,H are the circumcenter and orthocenter of △ABC\triangle ABC respectively,GG is the midpoint of segment AHAH , BEBE is the altitude on ACAC . Prove that if OE∥BCOE\parallel BC, then HH is the incenter of △GBC\triangle GBC.
geometry
China South East Mathematical Olympiad 2021 Grade11 P2

Source:

7/28/2021
Let p≥5p\geq 5 be a prime number, and set M={1,2,⋯ ,p−1}.M=\{1,2,\cdots,p-1\}. Define T={(n,xn):p∣nxn−1 and n,xn∈M}.T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}. If ∑(n,xn)∈Tn[nxnp]≡k(modp),\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p}, with 0≤k≤p−1,0\leq k\leq p-1, where [α]\left[\alpha\right] denotes the largest integer that does not exceed α,\alpha, determine the value of k.k.
number theoryprime numbers