6th Problem of The Week September 7 1. Calculate

1 1 1 + + ··· + . 1+2 1+2+3 1 + 2 + · · · + 51

2. Find the smallest value of 3a2 + 27b2 + 5c2 − 18ab − 30c + 237, where a, b, c are real numbers. 3. In parallelogram ABCD, DA is extended through A to point P and P C intersects AB, BD at Q, R respectively. If P Q = 735 and QR = 112, find RC. 4. Compute (104 + 324)(224 + 324)(344 + 324)(464 + 324)(584 + 324) (44 + 324)(164 + 324)(284 + 324)(404 + 324)(524 + 324) 7th Problem of The Week September 21 1. Let N be a number such that its cube root is equal to the square root of the sum of its digits. How many factors does N have? 2. Find a positive integer x such that x(x + 1)(x + 2)(x + 3) + 1 = 3792 . 3. Convex quadrilateral ABCD has BC = 8, CD = 12, AD = 10 and √ 6 A = 6 B = 60◦ . If AB = p + q, where p, q are positive integers, find p + q. 4. Right triangle ABC has 6 B = 90◦ and AC = 2. Point D lies on side AC such that 36 BAD = 6 BAC. If CD = 1, find BD.

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