3

Part of 2023 Czech-Polish-Slovak Junior Match

Problems(2)

Q symmetric of P wrt BH wanted, orthocenter related

Source: 2023 Czech-Polish-Slovak Match Junior, individual p3 CPSJ

Given is an acute triangle

ABC

P

lies inside this triangle and lies on the bisector of angle

\angle BAC

. Suppose that the point of intersection of the altitudes

H

ABP

lies inside triangle

ABC

Q

be the intersection of the line

AP

and the line perpendicular to

AC

passing through

H

Q

is the point symmetrical to

P

BH

geometryorthocenter

n persons a a party, everyone dislikes only 1

Source: 2023 Czech-Polish-Slovak Match Junior, Team p3 CPSJ

n

people met at the party, with

n \ge 2

. Each person dislikes exactly one other person present at the party (but not necessarily reciprocal, i.e. it may happen that

A

B

B

does not dislike

A

) and likes all others. Prove that guests can be seated at three tables in such a way that each guest likes all the people at his table.