# 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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