CMO 2023

Define the sequences $(a_n),(b_n)$ by \begin{align*} & a_n, b_n > 0, \forall n\in\mathbb{N_+} & a_{n+1} = a_n - \frac{1}{1+\sum_{i=1}^n\frac{1}{a_i}} & b_{n+1} = b_n + \frac{1}{1+\sum_{i=1}^n\frac{1}{b_i}} \end{align*}

1. If $a_{100}b_{100} = a_{101}b_{101}$ , find the value of $a_1-b_1$ ;
2. If $a_{100} = b_{99}$ , determine which is larger between $a_{100}+b_{100}$ and $a_{101}+b_{101}$ .

Let $\triangle ABC$ be an equilateral triangle of side length 1. Let $D,E,F$ be points on $BC,AC,AB$ respectively, such that $\frac{DE}{20} = \frac{EF}{22} = \frac{FD}{38}$ . Let $X,Y,Z$ be on lines $BC,CA,AB$ respectively, such that $XY\perp DE, YZ\perp EF, ZX\perp FD$ . Find all possible values of $\frac{1}{[DEF]} + \frac{1}{[XYZ]}$ .

Given positive integer $m,n$ , color the points of the regular $(2m+2n)$ -gon in black and white, $2m$ in black and $2n$ in white. The coloring distance $d(B,C)$ of two black points $B,C$ is defined as the smaller number of white points in the two paths linking the two black points. The coloring distance $d(W,X)$ of two white points $W,X$ is defined as the smaller number of black points in the two paths linking the two white points. We define the matching of black points $\mathcal{B}$ : label the $2m$ black points with $A_1,\cdots,A_m,B_1,\cdots,B_m$ satisfying no $A_iB_i$ intersects inside the gon. We define the matching of white points $\mathcal{W}$ : label the $2n$ white points with $C_1,\cdots,C_n,D_1,\cdots,D_n$ satisfying no $C_iD_i$ intersects inside the gon. We define $P(\mathcal{B})=\sum^m_{i=1}d(A_i,B_i), P(\mathcal{W} )=\sum^n_{j=1}d(C_j,D_j)$ . Prove that: $\max_{\mathcal{B}}P(\mathcal{B})=\max_{\mathcal{W}}P(\mathcal{W})$

Find the minimum positive integer $n\ge 3$ , such that there exist $n$ points $A_1,A_2,\cdots, A_n$ satisfying no three points are collinear and for any $1\le i\le n$ , there exist $1\le j \le n (j\neq i)$ , segment $A_jA_{j+1}$ pass through the midpoint of segment $A_iA_{i+1}$ , where $A_{n+1}=A_1$

Prove that there exist $C>0$ , which satisfies the following conclusion: For any infinite positive arithmetic integer sequence $a_1, a_2, a_3,\cdots$ , if the greatest common divisor of $a_1$ and $a_2$ is squarefree, then there exists a positive integer $m\le C\cdot {a_2}^2$ , such that $a_m$ is squarefree. Note: A positive integer $N$ is squarefree if it is not divisible by any square number greater than $1$ .

Proposed by Qu Zhenhua

There are $n(n\ge 8)$ airports, some of which have one-way direct routes between them. For any two airports $a$ and $b$ , there is at most one one-way direct route from $a$ to $b$ (there may be both one-way direct routes from $a$ to $b$ and from $b$ to $a$ ). For any set $A$ composed of airports $(1\le | A| \le n-1)$ , there are at least $4\cdot \min \{|A|,n-|A| \}$ one-way direct routes from the airport in $A$ to the airport not in $A$ . Prove that: For any airport $x$ , we can start from $x$ and return to the airport by no more than $\sqrt{2n}$ one-way direct routes.