$\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$...

1. $\odot O_1$ and $\odot O_2$ intersect at two points $A$, $B$. The extension of $O_1A$ meets $\odot O_2$ at $C$, and the extension of $O_2A$ meets $\odot O_1$ at $D$, and through $B$ draw $BE \parallel O_2A$ intersecting $\odot O_1$ again at $E$. If $DE \parallel O_1A$, prove that $DC \perp CO_2$.
2. Let $x_1,x_2,\ldots,x_n$ be real numbers, where $n\ge 2$ is a given integer, and let $\lfloor{x_1}\rfloor,\lfloor{x_2}\rfloor,\ldots,\lfloor{x_n}\rfloor$ be a permutation of $1,2,\ldots,n$. Find the maximum and minimum of $\sum\limits_{i=1}^{n-1}\lfloor{x_{i+1}-x_i}\rfloor$ (here $\lfloor x\rfloor$ is the largest integer not greater than $x$).
3. There are $n$ students; each student knows exactly $d$ girl students and $d$ boy students ("knowing" is a symmetric relation). Find all pairs $(n,d)$ of integers.
4. For an integer $m\geq 4,$ let $T_{m}$ denote the number of sequences $a_{1},\dots,a_{m}$ such that the following conditions hold
i) For all $i=1,2,\dots,m$ we have $a_{i}\in \{1,2,3,4\}$,
ii) $a_{1} = a_{m} = 1$ and $a_{2}\neq 1$,
iii) For all $i=3,4\cdots, m, a_{i}\neq a_{i-1}, a_{i}\neq a_{i-2}.$
Prove that there exists a geometric sequence of positive integers $\{g_{n}\}$ such that for $n\geq 4$ we have that $g_{n} - 2\sqrt{g_{n}} < T_{n} < g_{n} + 2\sqrt{g_{n}}.$
5. Let $a$ be a positive integer, but not a perfect square; $r$ is a real root of the equation $x^3-2ax+1=0$. Prove that $r+\sqrt{a}$ is an irrational number.
6. In acute triangle $ABC$, $AB > AC$. $D$ and $E$ are the midpoints of $AB$, $AC$ respectively. The circumcircle of $ADE$ intersects the circumcircle of $BCE$ again at $P$. The circumcircle of $ADE$ intersects the circumcircle $BCD$ again at $Q$. Prove that $AP = AQ$.
7. Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $$(f(A_{p})-f(A))(f(B_{p})-f(B)).$$
8. Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$. Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$. If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.

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