Problem 2. Let $M\subseteq\{1,2,\ldots,2011\}$ be a subset satisfying the following condition:
For any three elements in $M$ , there exist two of them $a$ and $b$ , such that $a\mid b$ or $b\mid a$ .
Determine the maximum value of $| M |$ , where $| M |$ denotes the number of elements of $M$ .
Problem 3. Let $n\ge 2$ be a given integer.
$(1)$ Prove that: One can arrange all the subsets of the set $\{1,2,\ldots,n\}$ as a sequence of subsets $A_1,A_2,\ldots, A_{2^n}$ , such that $|A_{i+1}|=|A_i| +1$ or $|A_i|-1$ , where $i=1,2,3,\ldots,2^n$ and $A_{2^n+1}=A_1$ ;
$(2)$ Determine all possible values of the sum
- $\displaystyle\sum\limits_{i=1}^{2^n}(-1)^iS(A_i)\;,$
Problem 4. As shown in the figure, $AB$ and $CD$ are two chords in the circle $\odot O,\,AB\ne CD.\;\odot I$ is tangent to $\odot O$ internally at point $F$, and is tangent to the chords $AB$ and $CD$ at points $G$ and $H$ respectively . $\ell$ is a line passing through $O$ , meeting $AB, CD$ at points $P,Q$ respectively, such that $EP=EQ$ . Line $EF$ meets the line $\ell$ at point $M$ . Prove that the line through $M$ and parallel to the line $AB$ is tangent to the circle $\odot O$ .
- Pr4_CWMO2011.png (17 KiB) Vizualizat de 2086 ori
Problem 6. Let $a,b,c>0$, prove that
$\displaystyle\frac{(a-b)^2}{(c+a)(c+b)}+\frac{(b-c)^2}{(a+b)(a+c)}+\frac{(c-a)^2}{(b+c)(b+a)}\ge \frac{(a-b)^2}{a^2+b^2+c^2}\;.$
Problem 7. As shown in the figure , $AB>AC$ , and the incircle $\odot I$ of $\triangle ABC$ is tangent to $BC,CA$ and $AB$ at points $D,E$ and $F$ respectively. Let $M$ be the midpoint of side $BC$ , and and $AH\perp BC$ at the point $H$ . The bisector $AI$ of $\angle BAC$ intersects the lines $DE,DF$ at points $K,L$ respectively.
Prove that $M,L,H$ and $K$ are concyclic.
- Pr7_CWMO2011.png (12.23 KiB) Vizualizat de 2086 ori