FKMO 2017

A acute triangle $\triangle ABC$ has circumcenter $O$ . The circumcircle of $OAB$ , called $O_1$ , and the circumcircle of $OAC$ , called $O_2$ , meets $BC$ again at $D ( \not=B )$ and $E ( \not= C )$ respectively. The perpendicular bisector of $BC$ hits $AC$ again at $F$ . Prove that the circumcenter of $\triangle ADE$ lies on $AC$ if and only if the centers of $O_1, O_2$ and $F$ are colinear.

For a positive integer $n$ , $(a_0, a_1, \cdots , a_n)$ is a $n+1$ -tuple with integer entries. For all $k=0, 1, \cdots , n$ , we denote $b_k$ as the number of $k$ s in $(a_0, a_1, \cdots ,a_n)$ . For all $k = 0,1, \cdots , n$ , we denote $c_k$ as the number of $k$ s in $(b_0, b_1, \cdots ,b_n)$ . Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$ , $a_1=c_1$ , $\cdots$ , $a_n=c_n$ .

For a positive integer $n$ , denote $c_n=2017^n$ . A function $f: \mathbb{N} \rightarrow \mathbb{R}$ satisfies the following two conditions.

1. For all positive integers $m, n$ , $f(m+n) \le 2017 \cdot f(m) \cdot f(n+325)$ .

2. For all positive integer $n$ , we have $0<f(c_{n+1})<f(c_n)^{2017}$ .

Prove that there exists a sequence $a_1, a_2, \cdots$ which satisfies the following. For all $n, k$ which satisfies $a_k<n$ , we have $f(n)^{c_k} < f(c_k)^n$ .

For a positive integer $n \ge 2$ , define a sequence $a_1, a_2, \cdots ,a_n$ as the following. $$a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$ (a) Show that $a_1, a_2, \cdots a_n$ are all integers.

(b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.

Let there be cyclic quadrilateral $ABCD$ with $L$ as the midpoint of $AB$ and $M$ as the midpoint of $CD$ . Let $AC \cap BD = E$ , and let rays $AB$ and $DC$ meet again at $F$ . Let $LM \cap DE = P$ . Let $Q$ be the foot of the perpendicular from $P$ to $EM$ . If the orthocenter of $\triangle FLM$ is $E$ , prove the following equality. $$\frac{EP^2}{EQ} = \frac{1}{2} \left( \frac{BD^2}{DF} - \frac{BC^2}{CF} \right)$$

A room has $2017$ boxes in a circle. A set of boxes is friendly if there are at least two boxes in the set, and for each boxes in the set, if we go clockwise starting from the box, we would pass either $0$ or odd number of boxes before encountering a new box in the set. $30$ students enter the room and picks a set of boxes so that the set is friendly, and each students puts a letter inside all of the boxes that he/she chose. If the set of the boxes which have $30$ letters inside is not friendly, show that there exists two students $A, B$ and boxes $a, b$ satisfying the following condition.

(i). $A$ chose $a$ but not $b$ , and $B$ chose $b$ but not $a$ .

(ii). Starting from $a$ and going clockwise to $b$ , the number of boxes that we pass through, not including $a$ and $b$ , is not an odd number, and none of $A$ or $B$ chose such boxes that we passed.