3

Part of 2018 Korea Winter Program Practice Test

Problems(2)

Concyclic 6 points with simson lines

Source: 2018 Korea Winter Program Practice Test 1 #3

A_{DE}

by the foot of perpendicular line from

A

DE

. Given concyclic points

A,B,C,D,E,F

, show that the three points determined by the lines

A_{FD}A_{DE}

B_{DE}B_{EF}

C_{EF}C_{FD}

, and the three points determined by the lines

D_{CA}D_{AB}

E_{AB}E_{BC}

F_{BC}F_{CA}

Simson lineConcyclicgeometry

A set of Good Numbers

Source: 2018 Korea Winter Program Practice Test 1 #7

n

be a "Good Number" if sum of all divisors of

n

2n

n\in \mathbb{Z}.

Does there exist an infinite set

M

that satisfies the following?
For all

a,b\in M,

a+b

is good number. (

a=b