MathDB

1

Part of 2023 OMpD

Problems(2)

easy functional equation

Source: 2023 4th OMpD L3 P1 - Brazil - Olimpíada Matemáticos por Diversão

9/21/2023
Determine all functions f:Rā†’Rf : \mathbb{R} \rightarrow \mathbb{R} such that, for all real numbers xx and yy, f(x)(x+f(f(y)))=f(x2)+xf(y)f(x)(x+f(f(y))) = f(x^2)+xf(y)
functionfunctional equationreal numberalgebra
Easy tournament problem

Source: 2023 4th OMpD L2 P1 - Brazil - Olimpíada Matemáticos por Diversão

9/21/2023
Some friends formed 66 football teams, and decided to hold a tournament where each team faces each other exactly once in a match. In each match, whoever wins gets 33 points, whoever loses gets no points, and if the two teams draw, each gets 11 point.
At the end of the tournament, it was found that the teams' scores were 1010, 99, 66, 66, 44 and 22 points. Regarding this tournament, answer the following items, justifying your answer in each one.
(a) How many matches ended in a draw in the tournament?
(b) Determine, for each of the 66 teams, the number of wins, draws and losses.
(c) If we consider only the matches played between the team that scored 99 points against the two teams that scored 66 points, and the one played between the two teams that scored 66 points, explain why among these three matches, there are at least 22 draws.
combinatoricsTournament graphs