MathDB

3

Part of 2018 China Northern MO

Problems(2)

Divisibility by a prime

Source: 2018 China North Mathematical Olympiad Grade 10 Test 1 P3

7/26/2018
Let pp be a prime such that 3∣p+13|p+1. Show that p∣a−bp|a-b if and only if p∣a3−b3p|a^3-b^3
number theoryChina
Five points are concyclic

Source: 2018 China North Mathematical Olympiad Grade 11 Test 1 P3

6/24/2019
A,B,C,D,EA,B,C,D,E lie on ⊙O\odot O in that order,and BD∩CE=F,CE∩AD=G,AD∩BE=H,BE∩AC=I,AC∩BD=J.BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J. Prove that FGCE=GHDA=HIBE=IJAC=JFBD\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD} when and only when F,G,H,I,JF,G,H,I,J are concyclic.
geometry