MathDB

5

Part of 2017 239 Open Mathematical Olympiad

Problems(2)

Number of pairs having a mutual enemy is not less than number of enemies.

Source: 239 2017 J8 S5

5/22/2020
A school has three classes. Some pairs of children from different classes are enemies (there are no enemies in a class). It is known that every child from the first class has as many enemies in the second class as in the third; the same is true for other classes. Prove that the number of pairs of children from classes having a common enemy is not less than the number of pairs of children being enemies.
graph theorycombinatorics
geometry with length condition \sqrt{2}(BC_BA)=AC

Source: 239 2017 J5

6/3/2020
Given a quadrilateral ABCDABCD in which2(BC−BA)=AC.\sqrt{2}(BC-BA)=AC.Let XX be the midpoint of ACAC. Prove that 2∠BXD=∠DAB−∠DCB.2\angle BXD=\angle DAB - \angle DCB.
geometry