Solve the equation y 3 x 3 8x 2 6x 8 for positive integers x and y

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Previous 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. Next Question: 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.

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Solve the equation y^3 = x^3 + 8x^2 - 6x + 8, for positive integers x and y.: Asked In Maths Olympiad MAN (10 years ago); Unsolved; Is this Puzzle helpful? (0) (0) Submit Your Solution

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Latest Maths Olympiad Aptitude Question SOLUTION: Solve the equation y^3 = x^3 + 8x^2 - 6x + 8, for positive integers x and y.