These should take about 30 minutes for people who are beginners to high school maths olympiads.
Q1: Prove that (a^2/sqrt(b)+b^2/sqrt(c)+c^2/sqrt(a))*(1/a+1/b+1/c) is at least 9 when a, b, and c are all at least 1.
Q2: Find the area of the orthic triangle of ABC given ABC's side lengths.
Q3: If f(x) is a continuous one-to-one function, and for all x and y (f(f(x))+f(f(y)))/2=f((x+y)/2), find f(x)
HINTS:
Q1: AM-GM and then GM-HM
Q2: Heron's and 1/2absinC
Q3: What if x=y?
These should take about 30 minutes for people who are beginners to high school maths olympiads.
Q1: Prove that (a^2/sqrt(b)+b^2/sqrt(c)+c^2/sqrt(a))*(1/a+1/b+1/c) is at least 9 when a, b, and c are all at least 1.
Q2: Find the area of the orthic triangle of ABC given ABC's side lengths.
Q3: If f(x) is a continuous one-to-one function, and for all x and y (f(f(x))+f(f(y)))/2=f((x+y)/2), find f(x)