MathDB

13

Part of 2019 AIME Problems

Problems(2)

Circumcircle Intersections

Source: 2019 AIME I #13

3/14/2019
Triangle ABCABC has side lengths AB=4AB=4, BC=5BC=5, and CA=6CA=6. Points DD and EE are on ray ABAB with AB<AD<AEAB<AD<AE. The point F≠CF \neq C is a point of intersection of the circumcircles of △ACD\triangle ACD and △EBC\triangle EBC satisfying DF=2DF=2 and EF=7EF=7. Then BEBE can be expressed as a+bcd\tfrac{a+b\sqrt{c}}{d}, where aa, bb, cc, and dd are positive integers such that aa and dd are relatively prime, and cc is not divisible by the square of any prime. Find a+b+c+da+b+c+d.
AIMEAIME Igeometrycircumcircle2019 AIME Itrigonometrypower of a point
Point Inside an Octagon

Source: 2019 AIME II #13

3/22/2019
Regular octagon A1A2A3A4A5A6A7A8A_1A_2A_3A_4A_5A_6A_7A_8 is inscribed in a circle of area 11. Point PP lies inside the circle so that the region bounded by PA1‾\overline{PA_1}, PA2‾\overline{PA_2}, and the minor arc A1A2^\widehat{A_1A_2} of the circle has area 17\tfrac17, while the region bounded by PA3‾\overline{PA_3}, PA4‾\overline{PA_4}, and the minor arc A3A4^\widehat{A_3A_4} of the circle has area 19\tfrac 19. There is a positive integer nn such that the area of the region bounded by PA6‾\overline{PA_6}, PA7‾\overline{PA_7}, and the minor arc A6A7^\widehat{A_6A_7} is equal to 18−2n\tfrac18 - \tfrac{\sqrt 2}n. Find nn.
AMCAIMEAIME IIgeometry