MathDB

2

Part of 2001 CentroAmerican

Problems(2)

circle problem

Source: Central American Olympiad 2001, problem 2

8/12/2009
Let AB AB be the diameter of a circle with a center O O and radius 1 1. Let C C and D D be two points on the circle such that AC AC and BD BD intersect at a point Q Q situated inside of the circle, and \angle AQB\equal{} 2 \angle COD. Let P P be a point that intersects the tangents to the circle that pass through the points C C and D D. Determine the length of segment OP OP.
Algebra.

Source: Central American Olympiad 2001, problem 5

8/12/2009
Let a,b a,b and c c real numbers such that the equation ax^2\plus{}bx\plus{}c\equal{}0 has two distinct real solutions p1,p2 p_1,p_2 and the equation cx^2\plus{}bx\plus{}a\equal{}0 has two distinct real solutions q1,q2 q_1,q_2. We know that the numbers p1,q1,p2,q2 p_1,q_1,p_2,q_2 in that order, form an arithmetic progression. Show that a\plus{}c\equal{}0.
quadraticsfunctionarithmetic sequenceabsolute value