Math problems invented on 09/10/2012:
61. Given a ruler and a calculator just to be used to find the sine values of angles, show how to divide an angle into five equal smaller ones.
62. Let the bisectors of the exterior and interior angles of point A of triangle ABC intersect side BC at D and E, respectively. The circumcircle of triangle ADM where M is the midpoint of BC intersects AB and AC at I and J, respectively. IJ intersects BC at K; P and N are the midpoints of EK and IJ, respectively. Prove that E is also the incenter of triangle AHM.
63. 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 OPis the bisector of ∠EPF.
64. 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.
65. 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 B, P and L are collinear.
Silicon Valley Math University 