$AB$ is a chord of a circle with center $O$, $M$ is the midpoint of $AB$. A non-diameter chord is drawn through $M$ and intersects the circle at $C$ and $D$. The tangents of the circle from points $C$ and $D$ intersect line $AB$ at $P$ and $Q$, respectively. Prove that $PA$ = $QB$.

Let $f$ be a function from R to R. Suppose we have: (1) $f(0)=0$ (2) For all $x, y \in (-\infty, -1) \cup (1, \infty)$, we have $f(\frac{1}{x})+f(\frac{1}{y})=f(\frac{x+y}{1+xy})$. (3) If $x \in (-1,0)$, then $f(x) > 0$. Prove: $\sum_{n=1}^{+\infty} f(\frac{1}{n^2+7n+11}) > f(\frac12)$ with $n \in N^+$.

Let positive numbers $a_1, a_2, ..., a_{3n}$ $(n \geq 2)$ constitute an arithmetic progression with common difference $d > 0$. Prove that among any $n + 2$ terms in this progression, there exist two terms $a_i, a_j$ $(i \neq j)$ satisfying $1 < \frac{|a_i - a_j|}{nd} < 2$.

Let $A$ be the set of $n$-digit integers whose digits are all from $\{ 1, 2, 3, 4, 5 \}$. $B$ is subset of $A$ such that it contains digit $5$, and there is no digit $3$ in front of digit $5$ (i.e. for $n = 2$, $35$ is not allowed, but $53$ is allowed). How many elements does set $B$ have?

Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.

Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$