Math problems invented on 09/08/2012:
40. Given a triangle ABC with circumcircle C and circumcenter O, AB > AC > BC and ∠BAC = 30°. Let D be a point on the minor arc BC of the circumcircle, E and F be points on AD such that AB⊥OE and AC⊥OF. The segments BE and CF intersect at P. Prove that OP is the bisector of ∠EPF.
41. Given a triangle ABC with circumcircle C and circumcenter O, AB > AC > BC and ∠BAC = 30°. Let D be a point on the minor arc BC of the circumcircle, E and F be points on AD such that AB⊥OE and AC⊥OF. The segments BE and CF intersect at P. Let K be a point on BP such that PK = CK. Prove that BK = OP.
42. Given a triangle ABC with circumcircle C and circumcenter O, AB > AC > BC and ∠BAC = 30°. Let D be a point on the minor arc BC of the circumcircle, E and F be points on AD such that AB⊥OE and AC⊥OF. The segments BE and CF intersect at P. Draw circle C1 with center P and radius CP to meet C at L. Prove that the three points B, P and L are collinear.
Silicon Valley Math University 