CWMO 2014

Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$ .

Let $AB$ be the diameter of semicircle $O$ , $C, D$ be points on the arc $AB$ , $P, Q$ be respectively the circumcenter of $\triangle OAC$ and $\triangle OBD$ . Prove that: $CP\cdot CQ=DP \cdot DQ$ .

import cse5; import olympiad; unitsize(3.5cm); dotfactor=4; pathpen=black;
real h=sqrt(55/64);
pair A=(-1,0), O=origin, B=(1,0),C=shift(-3/8,h)*O,D=shift(4/5,3/5)*O,P=circumcenter(O,A,C), Q=circumcenter(O,D,B);
D(arc(O,1,0,180),darkgreen);
D(MP("A",A,W)--MP("C",C,N)--MP("P",P,SE)--MP("D",D,E)--MP("Q",Q,E)--C--MP("O",O,S)--D--MP("B",B,E)--cycle,deepblue);
D(O);

Attachments:

Let $A_1,A_2,...$ be a sequence of sets such that for any positive integer $i$ , there are only finitely many values of $j$ such that $A_j\subseteq A_i$ . Prove that there is a sequence of positive integers $a_1,a_2,...$ such that for any pair $(i,j)$ to have $a_i\mid a_j\iff A_i\subseteq A_j$ .

Given a positive integer $n$ , let $a_1,a_2,..,a_n$ be a sequence of nonnegative integers. A sequence of one or more consecutive terms of $a_1,a_2,..,a_n$ is called $dragon$ if their aritmetic mean is larger than 1. If a sequence is a $dragon$ , then its first term is the $head$ and the last term is the $tail$ . Suppose $a_1,a_2,..,a_n$ is the $head$ or/and $tail$ of some $dragon$ sequence; determine the minimum value of $a_1+a_2+\cdots +a_n$ in terms of $n$ .

Given a positive integer $m$ , Prove that there exists a positive integers $n_0$ such that all first digits after the decimal points of $\sqrt{n^2+817n+m}$ in decimal representation are equal, for all integers $n>n_0$ .

Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n$ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0$ , $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$ . Find the maximum of Min $\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$ .

In the plane, Point $O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$ . Prove that$$|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.$$

Given a real number $q$ , $1 < q < 2$ define a sequence $\{x_n\}$ as follows: for any positive integer $n$ , let $$x_n=a_0+a_1 \cdot 2+ a_2 \cdot 2^2 + \cdots + a_k \cdot 2^k (a_i \in \{0,1\}, i = 0,1, \cdots m k)$$ be its binary representation, define $$x_k= a_0 +a_1 \cdot q + a_2 \cdot q^2 + \cdots +a_k \cdot q^k.$$ Prove that for any positive integer $n$ , there exists a positive integer $m$ such that $x_n < x_m \leq x_n+1$ .