Let $x_i \in \{0, 1\} (i = 1, 2, \cdots, n)$. If the function $f = f(x_1, x_2, \cdots, x_n)$ only equals $0$ or $1$, then define $f$ as an "$n$-variable Boolean function" and denote $$D_n (f) = \{ (x_1, x_2, \cdots, x_n) | f(x_1, x_2, \cdots, x_n) = 0 \}$$. $(1)$ Determine the number of $n$-variable Boolean functions; $(2)$ Let $g$ be a $10$-variable Boolean function satisfying $$g(x_1, x_2, \cdots, x_{10}) \equiv 1 + x_1 + x_1 x_2 + x_1 x_2 x_3 + \cdots + x_1 x_2\cdots x_{10} \pmod{2}$$Evaluate the size of the set $D_{10} (g)$ and $\sum\limits_{(x_1, x_2, \cdots, x_{10}) \in D_{10} (g)} (x_1 + x_2 + x_3 + \cdots + x_{10})$.

Let $ABC$ be an acute-angled triangle. In $ABC$, $AB \neq AB$, $K$ is the midpoint of the the median $AD$, $DE \perp AB$ at $E$, $DF \perp AC$ at $F$. The lines $KE$, $KF$ intersect the line $BC$ at $M$, $N$, respectively. The circumcenters of $\triangle DEM$, $\triangle DFN$ are $O_1, O_2$, respectively. Prove that $O_1 O_2 \parallel BC$.

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 1, 2$. Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$.

Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M,N$ are the midpoint of arc $ADC,ABC$. $DO$ and $AN$ intersect each other at $G$, the line passes through $G$ and parellel to $NC$ intersect $CD$ at $K$. Prove that $AK\perp BM$.

The sequence $\{a_n\}$ satisfies $a_1 = \frac{1}{2}$, $a_2 = \frac{3}{8}$, and $a_{n + 1}^2 + 3 a_n a_{n + 2} = 2 a_{n + 1} (a_n + a_{n + 2}) (n \in \mathbb{N^*})$. $(1)$ Determine the general formula of the sequence $\{a_n\}$; $(2)$ Prove that for any positive integer $n$, there is $0 < a_n < \frac{1}{\sqrt{2n + 1}}$.

Let $m$ be a given positive integer. Define $a_k=\frac{(2km)!}{3^{(k-1)m}},k=1,2,\cdots.$ Prove that there are infinite many integers and infinite many non-integers in the sequence $\{a_k\}$.

Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set $$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and $$(x_1, y_1), (x_1, y_2), (x_1, y_3), (x_2, y_2) \in A.$$Determine the largest possible number of elements in $A$.

Let $x_i \in \{0,1\}(i=1,2,\cdots ,n)$,if the value of function $f=f(x_1,x_2, \cdots ,x_n)$ can only be $0$ or $1$,then we call $f$ a $n$-var Boole function,and we denote $D_n(f)=\{(x_1,x_2, \cdots ,x_n)|f(x_1,x_2, \cdots ,x_n)=0\}.$ $(1)$ Find the number of $n$-var Boole function; $(2)$ Let $g$ be a $n$-var Boole function such that $g(x_1,x_2, \cdots ,x_n) \equiv 1+x_1+x_1x_2+x_1x_2x_3 +\cdots +x_1x_2 \cdots x_n \pmod 2$, Find the number of elements of the set $D_n(g)$,and find the maximum of $n \in \mathbb{N}_+$ such that $\sum_{(x_1,x_2, \cdots ,x_n) \in D_n(g)}(x_1+x_2+ \cdots +x_n) \le 2017.$

Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$

For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set $$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$. Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.

Let $a, b, c$ be real numbers, $a \neq 0$. If the equation $2ax^2 + bx + c = 0$ has real root on the interval $[-1, 1]$. Prove that $$\min \{c, a + c + 1\} \leq \max \{|b - a + 1|, |b + a - 1|\},$$and determine the necessary and sufficient conditions of $a, b, c$ for the equality case to be achieved.

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $O$, where $AC\perp BD$. $M$ be the midpoint of arc $ADC$. Circle $(DOM)$ intersect $DA,DC$ at $E,F$. Prove that $BE=BF$.

Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$

Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set $$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, x_3, y_1, y_2, y_3$ such that $x_1 < x_2 < x_3, y_1 < y_2 < y_3$ and $$(x_1, y_2), (x_2, y_1), (x_2, y_2), (x_2, y_3), (x_3, y_2) \in A.$$Determine the largest possible number of elements in $A$.