6

Part of 2018 Brazil National Olympiad

Problems(2)

Points and triangles in the plane

Source: Brazilian Mathematical Olympiad 2018 - Q6

4n

points in the plane, with no three points collinear. Using these points as vertices, we form

\binom{4n}{3}

triangles. Show that there exists a point

X

of the plane that belongs to the interior of at least

2n^3

of these triangles.

combinatorial geometrycombinatoricsBrazilian Math OlympiadBrazilian Math Olympiad 2018geometry

Source: Brazil MO 2018 Grades 8 and 9

S(n)

be the sum of digits of

n

. Determine all the pairs

(a, b)

of positive integers, such that the expression

S(an + b) - S(n)

has a finite number of values, where

n

is varying in the positive integers.