MathDB

2

Part of 2017 Irish Math Olympiad

Problems(2)

a + b + c = 0 , a^2 + b^2 + c^2 = 1 , a^3 + b^3 +c^3 = 4abc

Source: Irish MO 2017 paper 1 problem 2

12/12/2022
Solve the equations : {a+b+c=0a2+b2+c2=1a3+b3+c3=4abc\begin{cases} a + b + c = 0 \\ a^2 + b^2 + c^2 = 1\\a^3 + b^3 +c^3 = 4abc \end{cases} for a,b, a,b, and c.c.
algebrasystem of equationsSystem
5 teams play in a soccer competition

Source: Irish MO 2017 paper 2 problem 2

12/12/2022
55 teams play in a soccer competition where each team plays one match against each of the other four teams. A winning team gains 55 points and a losing team 00 points. For a 0−00-0 draw both teams gain 11 point, and for other draws (1−1,2−2,3−3,1-1,2-2,3-3,etc.) both teams gain 2 points. At the end of the competition, we write down the total points for each team, and we find that they form 5 consecutive integers. What is the minimum number of goals scored?
combinatorics