USAJMO 2012 questions

  1. Given a triangle ABC, let P and Q be the points on the segments AB and AC, respectively such that AP = AQ. Let S and R be distinct points on segment BC such that S lies between B and R, ∠BPS = ∠PRS, and ∠CQR = ∠QSR. Prove that P, Q, R and S are concyclic (in other words these four points lie on a circle).
  2. Find all integers \((n \ge 3 ) \) such that among any n positive real numbers \(( a_1 , a_2 , … , a_n ) \) with \(\displaystyle {\text(\max)(a_1 , a_2 , … , a_n) \le n) (\min)(a_1 , a_2 , … , a_n)} \) there exist three that are the side lengths of an acute triangle.
  3. Let a, b, c be positive real numbers. Prove that \(\displaystyle {(\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2))} \).
  4. Let \((\alpha)\) be an irrational number with \((0 < \alpha < 1)\), and draw a circle in the plane whose circumference has length 1. Given any integer \((n \ge 3 )\), define a sequence of points \((P_1 , P_2 , … , P_n )\) as follows. First select any point \((P_1)\) on the circle, and for \(( 2 \le k \le n ) \) define \((P_k)\) as the point on the circle for which the length of the arc \((P_{k-1} P_k)\) is \((\alpha)\), when travelling counterclockwise around the circle from \((P_{k-1} )\) to \((P_k)\). Suppose that \((P_a)\) and \((P_b)\) are the nearest adjacent points on either side of \((P_n)\). Prove that \((a+b \le n)\).
  5. For distinct positive integers a, b < 2012, define f(a, b) to be the number of integers k with (1le k < 2012) such that the remainder when ak divided by 2012 is greater than that of bk divided by 2012. Let S be the minimum value of f(a, b), where a and b range over all pairs of distinct positive integers less than 2012. Determine S.
  6. Let P be a point in the plane of triangle ABC, and \((\gamma)\) be a line passing through P. Let A’, B’, C’  be the points where reflections of the lines PA, PB, PC with respect to \((\gamma)\) intersect lines BC, AC, AB, respectively. Prove that A’, B’ and C’ are collinear.

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