Bay Area Mathematical Olympiad #0ij6

Bay Area Mathematical OlympiadHard

Community difficultyNo ratings yetBe the first to rate this problem.

Understand the scale

Scale 0.5: Easiest math competition problems, often solvable even for early middle school students without experience. 1: Problems strictly for beginners, often at a moderate middle school level. 1.5: Problems for stronger beginner students, on the level of the middling problems in most middle school contests. 2: For motivated beginners, harder questions from the previous categories. 2.5: More advanced beginner problems, hardest questions from previous categories. 3: Early intermediate problems that require more creative thinking. 3.5: Tougher early intermediate problems that consistently stretch into higher-level creative thinking or conceptual knowledge. 4: Intermediate-level problems. 4.5: Upper intermediate problems approaching the upper bound for non-invitational sprint math competitions. 5: More difficult AIME problems or simple proof-based Olympiad-style problems. 6: High-level AIME-styled questions or introductory Olympiad-level questions. 7: Tougher Olympiad-level questions, may require more technical knowledge. 8: High-level Olympiad-level questions. 9: Expert Olympiad-level questions. 9.5: The hardest problems appearing on Olympiads which the strongest students could reasonably solve. 10: Historically notorious problems, generally unsuitable for even very hard competitions because they are exceedingly tedious, long, and difficult.

Triangle centers: centroid, incenter, circumcenter, orthocenter, Euler line, nine-point circleMenelaus' theoremHomothetyAngle chasingConstructions and loci

Problem

Problem:

In triangle $A B C$ , choose point $A_{1}$ on side $B C$ , point $B_{1}$ on side $C A$ , and point $C_{1}$ on side $A B$ in such a way that the three segments $A A_{1}$ , $B B_{1}$ , and $C C_{1}$ intersect in one point $P$ . Prove that $P$ is the centroid of triangle $A B C$ if and only if $P$ is the centroid of triangle $A_{1} B_{1} C_{1}$ .

← Back to all problems