Unizor - Combinatorics - Advanced Problems 1.5.
Channel: Zor Shekhtman on UNIZOR Education
Date: 04/06/2014

Problem 1
Consider N straight lines on the plane. Intersecting in some way, they divide the entire plane into separate parts.
What is the maximum number of parts the plane can be divided into by these N lines?
Solution
Answer: 1 + N·(N+1)/2

Problem 2
Consider N circles on the plane, possibly of different radiuses. Intersecting in some way, they divide the entire plane into separate parts.
What is the maximum number of parts the plane can be divided into by these N circles?
Solution
Answer: 2 + N·(N−1)

Problem 3
Solve the equation
(C(4,x))^(−1) − (C(5,x))^(−1) = (C(6,x))^(−1)
Solution
Answer: x = 2

Problem 4
How many acute angles are formed by N rays originated from the same point on a plane, assuming that any two angles that do not coincide are different, even if their measures are the same?
Solution
Answer: C(N,2)