Subcontests
(4)If it passes through the baricentre, it has some cool properties!
ABCD is a scalene tetrahedron and let G be its baricentre. A plane α passes through G such that it intersects neither the interior of ΔBCD nor its perimeter. Prove that
dist(A,α)=dist(B,α)+dist(C,α)+dist(D,α). (Adapted from folklore) The gcd of 3 numbers instead of 2
Let n≥3 be an integer. Let s(n) be the number of (ordered) pairs (a;b) consisting of positive integers a,b from the set {1,2,…,n} which satisfy gcd(a,b,n)=1. Prove that s(n) is divisible by 4 for all n≥3. (Vlad Robu)