Maths Olympiad Exam

Find all integers a, b, c, d satisfying the following relations:
(i) 1 a b c d;
(ii) ab + cd = a + b + c + d + 3.

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Maths Olympiad Other Question

The circumference of a circle is divided into eight arcs by a convex quadrilateral ABCD, with
four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of
the arcs lying inside the quadrilateral are denoted by p, q, r, s in counter-clockwise direction
starting from some arc. Suppose p + r = q + s. Prove that ABCD is a cyclic quadrilateral. Let ABC be a triangle in which AB = AC and CAB = 90◦. Suppose M and
N are points on the hypotenuse BC such that BM2 + CN2 = MN2. Prove that
MAN = 45◦.