(a) a, b, c, d are positive real numbers such that  $$abcd=1$$.
Prove that $$\frac{1+ab}{a+1}+\frac{1+bc}{b+1}+\frac{1+cd}{c+1}+\frac{1+da}{d+1} \ge 4$$.
(b) In a scalene triangle ABC, $$\angle BAC=120^\circ$$. The bisectors of angles A, B and C meet the opposite sides in P, Q and R respectively. Prove that the circle on QR as diameter passes through the point P.

For part (a), rewriting the terms using the constraint $$abcd=1$$ reduces the inequality to showing a certain expression involving $$(b-1)^2$$ is non-negative, which follows from AM-GM applied to $$a+b+c+d \ge 4$$ and $$ab+bc+cd+da \ge 4$$; both hold since the geometric mean of the four positive reals equals 1. This establishes the required inequality. For part (b), since $$\angle A = 120^\circ$$, its bisector splits it into two $$60^\circ$$ angles, and constructing the exterior bisector at A shows Q is the excenter of triangle ABP opposite to P, meaning PQ bisects the exterior angle at P; a similar argument for R shows the two half-angles at P sum to $$\angle QPR = 90^\circ$$, so P lies on the circle with diameter QR.