In a triangle with angles $P$, $Q$, $R$, let $\alpha$, $\beta$, $\gamma$ be the angles $\angle QCR = 2P$, $\angle QIR = Q + R$, $\angle QOR = P + Q/2 + R/2$ respectively. Show that $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} > \frac{1}{45}$.

Consider a triangle $P Q R$ having sides of lengths $p , q$ and $r$ opposite to the angles $P , Q$ and $R$, respectively. Then which of the following statements is (are) TRUE ?
(A) $\cos P \geq 1 - \frac { p ^ { 2 } } { 2 q r }$
(B) $\cos R \geq \left( \frac { q - r } { p + q } \right) \cos P + \left( \frac { p - r } { p + q } \right) \cos Q$
(C) $\frac { q + r } { p } < 2 \frac { \sqrt { \sin Q \sin R } } { \sin P }$
(D) If $p < q$ and $p < r$, then $\cos Q > \frac { p } { r }$ and $\cos R > \frac { p } { q }$

For any positive integer $n \geq 2$, let $T_{n}$ denote a triangle with side lengths $n, n+1, n+2$. Select the correct options. (Note: If a triangle has side lengths $a, b, c$ respectively, let $s = \frac{a+b+c}{2}$, then the area of the triangle is $\sqrt{s(s-a)(s-b)(s-c)}$)
(1) $T_{n}$ is always an acute triangle
(2) The perimeters of $T_{2}, T_{3}, T_{4}, \cdots, T_{10}$ form an arithmetic sequence
(3) The area of $T_{n}$ increases as $n$ increases
(4) The three altitudes of $T_{5}$ form an arithmetic sequence in order
(5) The largest angle of $T_{3}$ is greater than the largest angle of $T_{2}$

On a plane, there is a triangle $A B C$ where $\angle A = 91 ^ { \circ }, \angle C = 29 ^ { \circ }$. Let $\overline { B C } = a, \overline { C A } = b, \overline { A B } = c$. Select the correct options.
(1) $a ^ { 2 } > b ^ { 2 } + c ^ { 2 }$
(2) $\frac { c } { a } > \sin 29 ^ { \circ }$
(3) $\frac { b } { a } > \cos 29 ^ { \circ }$
(4) $\frac { a ^ { 2 } + b ^ { 2 } - c ^ { 2 } } { a b } < \sqrt { 3 }$
(5) The circumradius of triangle $A B C$ is less than $c$

The lengths of the sides $Q R , R P$ and $P Q$ in triangle $P Q R$ are $a , a + d$ and $a + 2 d$ respectively, where $a$ and $d$ are positive and such that $3 d > 2 a$.
What is the full range, in degrees, of possible values for angle $P R Q$ ?
A $0 <$ angle $P R Q < 60$
B 0 < angle $P R Q < 120$
C 60 < angle $P R Q < 120$
D 60 < angle $P R Q < 180$
E 120 < angle $P R Q < 180$