Subcontests
(5)Convex quadrilaterals and equal ratios
Let P be the point of intersection of the diagonals of a convex quadrilateral ABCD.Let X,Y,Z be points on the interior of AB,BC,CD respectively such that XBAX=YCBY=ZDCZ=2. Suppose that XY is tangent to the circumcircle of △CYZ and that YZ is tangent to the circumcircle of △BXY.Show that ∠APD=∠XYZ. 4x4 system, ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6
Determine all 4-tuples (a,b,c,d) of real numbers satisfying the following four equations: ⎩⎨⎧ab+c+d=3bc+d+a=5cd+a+b=2da+b+c=6