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June 25th, 2014, 10:05 AM
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Solved Paper for Regional Maths Olympiad RMO
Provide me solved question paper for Regional Maths Olympiad RMO examination in PDF file format? Here I am giving you solved question paper for Regional Maths Olympiad RMO examination in PDF file attached with it so you can get it easily. 1. Let ABC be an acute angled triangle. The circle _ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine \BAC given that the orthocenter of triangle APQ lies on _. Solution. Let K denote the orthocenter of triangle APQ. Since triangles ABC and AQP are similar it follows that K lies in the interior of triangle APQ. Note that \KPA = \KQA = 90__\A. Since BPKQ is a cyclic quadrilateral it follows that \BQK = 180_ _\BPK = 90_ _\A, while on the other hand \BQK = \BQA_\KQA = \A since BQ is perpendicular to AC. This shows that 90_ _ \A = \A, so \A = 45_. 2. Let f(x) = x3 + ax2 + bx + c and g(x) = x3 + bx2 + cx + a, where a; b; c are integers with c 6= 0. Suppose that the following conditions hold: (a) f(1) = 0; (b) the roots of g(x) are squares of the roots of f(x). Find the value of a2013 + b2013 + c2013. Solution. Note that g(1) = f(1) = 0, so 1 is a root of both f(x) and g(x). Let p and q be the other two roots of f(x), so p2 and q2 are the other two roots of g(x). We then get pq = _c and p2q2 = _a, so a = _c2. Also, (_a)2 = (p+q+1)2 = p2+q2+1+2(pq+p+q) = _b+2b = b. Therefore b = c4. Since f(1) = 0 we therefore get 1 + c _ c2 + c4 = 0. Factorising, we get (c + 1)(c3 _ c2 + 1) = 0. Note that c3 _ c2 + 1 = 0 has no integer root and hence c = _1; b = 1; a = _1. Therefore a2013 + b2013 + c2013 = _1. Last edited by Neelurk; February 7th, 2020 at 10:56 AM. |
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