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Please answer the following questions of RMO:

Q1. Let ABC be an acute-angled triangle and suppose angle ABC is the largest angle of the triangle. let R be its circumcentre. Suppose the circumcircle of triangle ARB cuts BC again in X. Prove that RX is perpendicular to BC.

Q2. find all real numbers x and y such that

y(

Q3. prove that there does not exist any positive integer n < 2310 such that n(2310-n) is a multiple of 2310.

Q4. find all positive real numbers x, y, z suh that

2x- 2y + \frac{1}{z} = \frac{1}{2014} \\ \\ 2y- 2z + \frac{1}{x} = \frac{1}{2014} \\ \\ 2z- 2x + \frac{1}{y} = \frac{1}{2014}[/tex]

Q5. Let ABC be a triangle. Let X be on the segment BC such AB=AX. Le AX meet the circumcircle T of triangle ABC again at D. Show that the circumcentre of triangle BDX lies on T.

Q6. For any natural number n, let S(n) denote the sum of the digits of n. Find the number of all 3-digit numbers n such that S(S(n) = 2

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