Maths Olympiad Exam

Solve the equation y^3 = x^3 + 8x^2 - 6x + 8, for positive integers x and y.

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Maths Olympiad Other Question

Produce AP and CQ to meet at K. Observe that AKCR is a rhombus and BQKP is a parallelogram. Put AP = x,CQ = y. Then PK = BQ = y, KQ = PB = x and AR = RC = CK = KA = x + y. Using cosine rule in triangle PKQ, we get PQ^2 = x^2 + y^2 - 2xy cos120 = x^2 + y^2 + xy. Similarly cosine rule in triangle QCR gives QR^2 = y^2 +(x+y)^2 - 2xy cos60 = x^2 +y^2+xy and cosine rule in triangle PAR gives RP^2 = x^2 + (x + y)^2 - 2xy cos 60 = x^2 + y^2 + xy. It follows that PQ = QR = RP. Suppose hx1, x2, . . . , xn, . . .i is a sequence of positive real numbers such that x1 x2
x3 x^n , and for all n
x1^1+x4^2+x9^3+..... +xn^2
n 1.
Show that for all k the following inequality is satisfied:
x1^1+x2^2+x3^3+ ...... +xk^k
n 3.