FKMO 2019

There are $n$ cards such that for each $i=1,2, \cdots n$ , there are exactly one card labeled $i$ . Initially the cards are piled with increasing order from top to bottom. There are two operations:

• $A$ : One can take the top card of the pile and move it to the bottom;
• $B$ : One can remove the top card from the pile.

The operation $ABBABBABBABB \cdots$ is repeated until only one card gets left. Let $L(n)$ be the labeled number on the final pile. Find all integers $k$ such that $L(3k)=k$ .

For a rectangle $ABCD$ which is not a square, there is $O$ such that $O$ is on the perpendicular bisector of $BD$ and $O$ is in the interior of $\triangle BCD$ . Denote by $E$ and $F$ the second intersections of the circle centered at $O$ passing through $B, D$ and $AB, AD$ . $BF$ and $DE$ meets at $G$ , and $X, Y, Z$ are the foots of the perpendiculars from $G$ to $AB, BD, DA$ . $L, M, N$ are the foots of the perpendiculars from $O$ to $CD, BD, BC$ . $XY$ and $ML$ meets at $P$ , $YZ$ and $MN$ meets at $Q$ . Prove that $BP$ and $DQ$ are parallel.

Prove that there exist infinitely many positive integers $k$ such that the sequence $\{x_n\}$ satisfying $$x_1=1, x_2=k+2, x_{n+2}-(k+1)x_{n+1}+x_n=0(n \ge 0)$$ does not contain any prime number.

Let triangle $ABC$ be an acute scalene triangle with orthocenter $H$ . The foot of perpendicular from $A$ to $BC$ is $O$ , and denote $K,L$ by the midpoints of $AB, AC$ , respectively. For a point $D(\neq O,B,C)$ on segment $BC$ , let $E,F$ be the orthocenters of triangles $ABD, ACD$ , respectively, and denote $M,N$ by the midpoints of $DE,DF$ . The perpendicular line from $M$ to $KH$ cuts the perpendicular line from $N$ to $LH$ at $P$ . If $Q$ is the midpoint of $EF$ , and $S$ is the orthocenter of triangle $HPQ$ , then prove that as $D$ varies on $BC$ , the ratio $\frac{OS}{OH}$ , $\frac{OQ}{OP}$ remains constant.

Find all pairs $(p,q)$ such that the equation $$x^4+2px^2+qx+p^2-36=0$$ has exactly $4$ integer roots(counting multiplicity).

A sequence $\{x_n \}=x_0, x_1, x_2, \cdots$ satisfies $x_0=a(1\le a \le 2019, a \in \mathbb{R})$ , and $$x_{n+1}=\begin{cases}1+1009x_n &\ (x_n \le 2) 2021-x_n &\ (2<x_n \le 1010) 3031-2x_n &\ (1010<x\le 1011) 2020-x_n &\ (1011<x_n) \end{cases}$$ for each non-negative integer $n$ . If there exist some integer $k>1$ such that $x_k=a$ , call such minimum $k$ a fundamental period of $\{x_n \}$ . Find all integers which can be a fundamental period of some seqeunce; and for such minimal odd period $k(>1)$ , find all values of $x_0=a$ such that the fundamental period of $\{x_n \}$ equals $k$ .