MathDB

2

Part of 2020 JBMO TST of France

Problems(2)

Geometry

Source: First JBMO TST of France 2020, Problem 2

3/5/2020
Let ABCABC be a triangle and KK be its circumcircle. Let PP be the point of intersection of BCBC with tangent in AA to KK. Let DD and EE be the symmetrical points of BB and AA, respectively, from PP. Let K1K_1 be the circumcircle of triangle DACDAC and let K2K_2 the circumscribed circle of triangle APBAPB. We denote with FF the second intersection point of the circles K1K_1 and K2K_2 Then denote with GG the second intersection point of the circle K1K_1 with BFBF. Show that the lines BCBC and EGEG are parallel.
geometry
Number Theory

Source: France JBMO TST 2020 Test 2 P2

3/10/2020
a) Find the minimum positive integer kk so that for every positive integers (x,y)(x, y) , for which x/y2x/y^2 and y/x2y/x^2, then xy/(x+y)kxy/(x+y) ^k b) Find the minimum positive integer ll so that for every positive integers (x,y,z)(x, y, z) , for which x/y2x/y^2, y/z2y/z^2 and z/x2z/x^2, then xyz/(x+y+z)lxyz/(x+y+z)^l
number theory