CN-SMO 2021

In $\triangle ABC$ , $AB=AC>BC$ , point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively, $G$ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$ , then $H$ is the incenter of $\triangle GBC$ .

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Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots$ , and $n$ , and the permutation is $S$ . For a permutation , a "descending chain" refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$ ), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other points in the chain are called the "non-starting point of descent" . For example: there are two descending chains $5, 2$ and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$ , delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$ of all possible n-point permutations $S$ . (1) Find $A(5)$ . (2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$

To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$ . (1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$ ; (2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.