CMO 2013


1. Two circles K_1 and K_2 of different radii intersect at two points A and B, let C and D be two points on K_1 and K_2, respectively, such that A is the midpoint of the segment CD. The extension of DB meets K_1 at another point E, the extension of CB meets K_2 at another point F. Let l_1 and l_2 be the perpendicular bisectors of CD and EF, respectively.
(1) Show that l_1 and l_2 have a unique common point (denoted by P).
(2) Prove that the lengths of CA, AP and PE are the side lengths of a right triangle.

2. Find all nonempty sets S of integers such that 3m-2n\in S for all (not necessarily distinct) m,n\in S.

3. Find all positive real numbers t with the following property: there exists an infinite set X of real numbers such that the inequality

\displaystyle \max\{\mid x-(a-d)\mid, \mid y-a\mid, \mid z-(a+d)\mid\}>td

holds for all (not necessarily distinct) x,y,z\in X, all real numbers a and all positive real numbers d.

4. n \geqslant 2, there are n finite sets {A_1},{A_2},...,{A_n} satisfy that for any i,j \in \left\{ {1,2,...,n} \right\}, \left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right|. Try to find the  the minimum of \sum\limits_{i = 1}^n {\left| {{A_i}} \right|} .

5. For any positive integer n and 0 \leqslant i \leqslant n, denote C_n^i \equiv c(n,i)(\bmod 2), where c(n,i) \in \left\{ {0,1} \right\}, define f(n,q) = \sum\limits_{i = 0}^n {c(n,i){q^i}} m,n,q are positive integers,  and q + 1 \ne {2^\alpha } for any \alpha\in N.  Prove that if f(m,q)\left| {f(n,q)} \right., then f(m,r)\left| {f(n,r)} \right. for any positive integer r.

6. m,n are positive integers, find the minimum positive integer N  satisties that if S is a set of integers contains a complete residue system of m and \left| S \right| = N, then there exit a nonempty set A \subseteq S, that n\left| {\sum\limits_{x \in A} x } \right..

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