Problem 51 In triangle ABC with 6 B = 90◦ , points D and E are on sides AB and AC respectively√such that AE bisects 6 BAC and CD bisects 6 ACB. If AE = 9 and CD = 8 2, find AC. √ √ Problem 52 The equation 3 x + 3 20 − x = 2 has two roots and the smaller √ of these roots is in the form p − q, where p and q are positive integers. Find p + q. Problem 53 Point P lies inside triangle ABC with sidelengths AB = 6, BC = 8 and CA = 7. Points D, E and F lie on sides AB, BC and CA, respectively, such that DP k BC, EP k CA, F P k AB and DP = EP = F P . m If the length of DP is in the form , where m and n are relatively prime n positive integers, find m + n. Problem 54 The reciprocal of the sum of the roots of 1000x6 − 1900x5 − 1400x4 − 190x3 − 130x2 − 38x − 30 = 0 r is , where r and s are relatively prime positive integers, find r + s. s Problem 55 Find all integral solutions to the equation m2 + 2m = n4 + 20n3 + 104n2 + 40n + 2003 .
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