MathDB

3

Part of 2012 JBMO TST - Turkey

Problems(2)

Show that all lines pass through a fixed point

Source: Turkey JBMO Team Selection Test Problem 3

5/29/2012
Let [AB][AB] be a chord of the circle Γ\Gamma not passing through its center and let MM be the midpoint of [AB].[AB]. Let CC be a variable point on Γ\Gamma different from AA and BB and PP be the point of intersection of the tangent lines at AA of circumcircle of CAMCAM and at BB of circumcircle of CBM.CBM. Show that all CPCP lines pass through a fixed point.
geometrycircumcirclegeometry proposed
2-variable symmetric inequality

Source: Turkey JBMO Team Selection Test Problem 7

5/30/2012
Show that for all real numbers x,yx, y satisfying x+y≥0x+y \geq 0 (x2+y2)3≥32(x3+y3)(xy−x−y) (x^2+y^2)^3 \geq 32(x^3+y^3)(xy-x-y)
inequalitiesfunctioninequalities proposed