FKMO 2010

Given an arbitrary triangle $ABC$ , denote by $P,Q,R$ the intersections of the incircle with sides $BC, CA, AB$ respectively. Let the area of triangle $ABC$ be $T$ , and its perimeter $L$ . Prove that the inequality $$\left(\frac {AB}{PQ}\right)^3 +\left(\frac {BC}{QR}\right)^3 +\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}$$ holds.

Let $I$ be the incentre and $O$ the circumcentre of a given acute triangle $ABC$ . The incircle is tangent to $BC$ at $D$ . Assume that $\angle B < \angle C$ and the segments $AO$ and $HD$ are parallel, where $H$ is the orthocentre of triangle $ABC$ . Let the intersection of the line $OD$ and $AH$ be $E$ . If the midpoint of $CI$ is $F$ , prove that $E,F,I,O$ are concyclic.

There are $n$ websites $1,2,\ldots,n$ ( $n \geq 2$ ). If there is a link from website $i$ to $j$ , we can use this link so we can move website $i$ to $j$ . For all $i \in \left\{1,2,\ldots,n - 1 \right\}$ , there is a link from website $i$ to $i+1$ . Prove that we can add less or equal than $3(n - 1)\log_{2}(\log_{2} n)$ links so that for all integers $1 \leq i < j \leq n$ , starting with website $i$ , and using at most three links to website $j$ . (If we use a link, website's number should increase. For example, No.7 to 4 is impossible).

Sorry for my bad English.

Given is a trapezoid $ABCD$ where $AB$ and $CD$ are parallel, and $A,B,C,D$ are clockwise in this order. Let $\Gamma_1$ be the circle with center $A$ passing through $B$ , $\Gamma_2$ be the circle with center $C$ passing through $D$ . The intersection of line $BD$ and $\Gamma_1$ is $P$ $( \ne B,D)$ . Denote by $\Gamma$ the circle with diameter $PD$ , and let $\Gamma$ and $\Gamma_1$ meet at $X$ $( \ne P)$ . $\Gamma$ and $\Gamma_2$ meet at $Y$ . If the circumcircle of triangle $XBY$ and $\Gamma_2$ meet at $Q$ , prove that $B,D,Q$ are collinear.

On a circular table are sitting $2n$ people, equally spaced in between. $m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.

(i) One may give cookies only to people adjacent to himself. (ii) In order to give a cookie to one's neighbor, one must eat a cookie.

Select arbitrarily a person $A$ sitting on the table. Find the minimum value $m$ such that there is a strategy in which $A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.

An arbitrary prime $p$ is given. If an integer sequence $(n_1 , n_2 , \cdots , n_k )$ satisfying the conditions

• For all $i= 1, 2, \cdots , k$ , $n_i \geq \frac{p+1}{2}$
• For all $i= 1, 2, \cdots , k$ , $p^{n_i} - 1$ is divisible by $n_{i+1}$ , and $\frac{p^{n_i} - 1}{n_{i+1}}$ is coprime to $n_{i+1}$ . Let $n_{k+1} = n_1$ . exists not for $k=1$ , but exists for some $k \geq 2$ , then call the prime a good prime. Prove that a prime is good iff it is not $2$ .