Maths Olympiad Exam

If a, b, c, x are real numbers such that abc 6= 0 and
xb + (1 - x)ca=xc + (1 - x)ab=xa + (1 - x)bc
then prove that either a + b + c = 0 or a = b = c.

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Maths Olympiad Other Question

Solve for integers x, y, z:
x + y = 1 - z, x^3 + y^3 = 1 - z^2. In a convex quadrilateral PQRS, PQ = RS, (√3+1)QR = SP and RSP−SPQ =
30◦. Prove that
PQR − QRS = 90◦.