MathDB

3

Part of 2016 Vietnam National Olympiad

Problems(2)

A point is on a fixed line

Source: Vietnam Mathematical Olympiad 2016, Day 1, Problem 3

1/6/2016
Let ABCABC be an acute triange with B,CB,C fixed. Let DD be the midpoint of BCBC and E,FE,F be the foot of the perpendiculars from DD to AB,ACAB,AC, respectively. a) Let OO be the circumcenter of triangle ABCABC and M=EF∩AO,N=EF∩BCM=EF\cap AO, N=EF\cap BC. Prove that the circumcircle of triangle AMNAMN passes through a fixed point; b) Assume that tangents of the circumcircle of triangle AEFAEF at E,FE,F are intersecting at TT. Prove that TT is on a fixed line.
geometry proposedgeometry
VMO 2016 Problem 7

Source: VMO 2016

1/7/2016
a) Prove that if nn is an odd perfect number then nn has the following form n=psm2 n=p^sm^2 where pp is prime has form 4k+14k+1, ss is positive integers has form 4h+14h+1, and m∈Z+m\in\mathbb{Z}^+, mm is not divisible by pp. b) Find all n∈Z+n\in\mathbb{Z}^+, n>1n>1 such that n−1n-1 and n(n+1)2\frac{n(n+1)}{2} is perfect number
number theoryperfect number