Inmo_mock_paper_5_-_2019_dec

1. Let ABC be a triangle. D is a point on the side (BC). The line AD meets the circumcircle again at X. P is the foot of the perpendicular from X to AB, and Q is the foot of the perpendicular from X to AC. Show that the line P Q is a tangent to the circle on diameter XD if and only if AB = AC. 2. (2004 IMO Shortlisted Problem) There are 10001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of k societies. Suppose the following conditions hold: (i) Each pair of students is in exactly one club. (ii) For each student and each society, the student is in exactly one club of the society. (iii) Each club has an odd number of students. In addition, a club with 2m+1 students (m is a positive integer) is in exactly m societies. Find all possible values of k. 3. In acute triangleABC, AB > AC. Let H be the foot of the perpendicular from A to BCand M be the midpoint of AH. Let D be the point where the incircle of ∆ABC is tangent to side BC. Let line DM intersect the incircle again at N . Prove that 6 BN D = 6 CN D. 4. Find all a ∈ R for which the functional equation f : R → R f (x − f (y)) = a(f (x) − x) − f (y) for all x, y ∈ R has a unique solution. 5. Let n be a positive integer. Find the number an of polynomials P (x) with coefficients in {0,1,2,3} such that P (2) = n. 6. Find the least integer n such that among every n distinct numbers a1 , a2 , ·· ·, an , chosen from [1,1000], there always exist ai , aj such that 0 < ai −aj < √ 1 + 3 3 ai aj .

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