Subcontests
(10)2018 JHMT Geometry #10
In an acute triangle ABC, the altitude from C intersects AB at E and the altitude from B intersects AC at D. CE and BD intersect at a point H. A circle with diameter DE intersects AB and AC at points F,G respectively. FG and AH intersect at K. If BC=25, BD=20, and BE=7, the length of AK is of the form qp , where p,q are relatively prime positive integers. Find p+q. 2018 JHMT Geometry #8
The vertical cross section of a circular cone with vertex P is an isoceles right triangle. Point A is on the base circle, point B is interior to the base circle, O is the center of the base circle, AB⊥OB at B, OH⊥PB at H, PA=4, and C is the midpoint of PA. When the volume of tetrahedron OHPC is maximized, the length of OB is x. x2 is in the form qp where p,q are relatively prime positive integers. Find p+q. 2018 JHMT Geometry #7
Let ℓ1, ℓ2, ℓ3m ℓ4 be rays from the origin, intersecting the line y=1 at x-coordinates −8,−2, 1, 2, respectively. For −81<k<21 , the line y=1+kx intersects the four rays at points A,B,C,D, respectively. When AB=CD, the ratio BCAB is qp where p,q are relatively prime positive integers. Find p+q. 2018 JHMT Geometry #4
Equilateral triangle OAB of side length 1 lies in the xy-plane (O is the origin). Let ℓ,m be the vertical lines passing through A,B, respectively. Let P,Q be on ℓ,m respectively such that the ratio OP:OQ:PQ=3:3:5. Let Q=(x,y,z). If z2=qp . where p,q are relatively prime positive integers, find p+q.