Regional Mathematics Olympiad (RMO) 2012

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
   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 \(\frac{a^3 + 3 b^3}{5a + b} + \frac{b^3 + 3c^3}{5b +c} + \frac{c^3 + 3a^3}{5c + a} \ge \frac{2}{3} (a^2 + b^2 + c^2)\).
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 \(1\le k < 2012\) 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.

Vietnam National Mathematical Olympiad 2012

Problem 1: Define a sequence as:
Prove that this sequence has a finite limit as Also determine the limit.

Problem 2:  Let and be two sequences of numbers, and let be an integer greater than Define Prove that if the quadratic expressions do not have any real roots, then all the remaining polynomials also don’t have real roots.

Problem 3:  Let be a cyclic quadrilateral with circumcentre and the pair of opposite sides not parallel with each other. Let and Denote, by the intersection of the angle bisectors of and and and Suppose that the four points are distinct.
(a) Show that the four points are concyclic. Find the centre of this circle, and denote it as
(b) Let Prove that are collinear.

Problem 4:  Let be a natural number. There are boys and girls standing in a line, in any arbitrary order. A student will be eligible for receiving candies, if we can choose two students of opposite sex with standing on either side of in ways. Show that the total number of candies does not exceed

Problem 5:  For a group of 5 girls, denoted as and boys. There are chairs arranged in a row. The students have been grouped to sit in the seats such that the following conditions are simultaneously met:
(a) Each chair has a proper seat.
(b) The order, from left to right, of the girls seating is
(c) Between and there are at least three boys.
(d) Between and there are at least one boy and most four boys.
How many such arrangements are possible?

Problem 6:  Consider two odd natural numbers and where is a divisor of and is a divisor of Prove that and are the terms of the series of natural numbers defined by

Problem 7:  Find all such that:
(a) For every real number there exist real number :
(b) If then
(c)