17. (This question is worth 12 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively, with $a = b \tan A$. (I) Prove that: $\sin \mathrm { B } = \cos \mathrm { A }$ (II) If $\sin C - \sin A \cos B = \frac { 3 } { 4 }$ and $B$ is an obtuse angle, find $A$, $B$, and $C$.

17. (This question is worth 12 points) Let the sides opposite to angles $A, B, C$ of $\triangle ABC$ be $a, b, c$ respectively. Given $a = b \tan A$ and $B$ is an obtuse angle. (I) Prove: $\mathrm { B } - \mathrm { A } = \frac { \pi } { 2 }$ (II) Find the range of $\sin \mathrm { A } + \sin \mathrm { C }$.

17. (12 points) In $\triangle \mathrm { ABC }$, the sides opposite to angles $\mathrm { A } , \mathrm { B }$, C are $a , b , c$ respectively. The vector $\vec { m } = ( a , \sqrt { 3 } b )$ is parallel to $\vec { n } = ( \cos \mathrm { A } , \sin \mathrm { B } )$. (I) Find A; (II) If $a = \sqrt { 7 } , b = 2$, find the area of $\triangle \mathrm { ABC }$.

22. (10 points) Elective 4-1: Geometric Proof As shown in the figure, AB is tangent to circle O at point B. Line AD intersects circle O at points D and E. $\mathrm { BC } \perp \mathrm { DE }$, with C as the foot of the perpendicular. (I) Prove that $\angle \mathrm { CBD } = \angle \mathrm { DBA }$; (II) If $\mathrm { AD } = 3 \mathrm { DC } , ~ \mathrm { BC }

17. (12 points) In $\triangle A B C$, the angles $A , B , C$ have opposite sides $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$.
(1) Find $A$;
(2) If $\sqrt { 2

17. (12 points) In $\triangle A B C$ , let the sides opposite to angles $A , B , C$ be $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$ .
(1) Find $A$ ;
(2) If $\sqrt { 2 } a + b = 2 c$ , find $\sin C$ .

18. (12 points) Let the sides opposite to angles $A$, $B$, $C$ of $\triangle A B C$ be $a$, $b$, $c$ respectively. Given that $\frac { \cos A } { 1 + \sin A } = \frac { \sin 2 B } { 1 + \cos 2 B }$.
(1) If $C = \frac { 2 \pi } { 3 }$, find $B$;
(2) Find the minimum value of $\frac { a ^ { 2 } + b ^ { 2 } } { c ^ { 2 } }$ .

125. In the figure below, chord $AB$ equals the radius of the circle and $AB \parallel CD$, angle $\beta = 2\alpha$, and $CX$ is tangent to the circle. How many degrees is $\widehat{BD}$?
\begin{minipage}{0.35\textwidth} [Figure: Circle with chord AB equal to radius, AB$\parallel$CD, tangent CX, angles $\alpha$ and $\beta$ marked] \end{minipage} \begin{minipage}{0.55\textwidth} (1) $50$
(2) $60$
(3) $70$
(4) $75$ \end{minipage}

4. Let $A B C$ be a triangle having $O$ and $I$ as its circumcentre and incentre respectively. If $R$ and $r$ are the circumradius and the inradius, respectively, then prove that (IO) $2 = R 2 - 2 R r$. Further show that the triangle BIO is a right-angled triangle if and only if $b$ is the arithmetic mean of a and c.

3. (a) In any triangle ABC , prove that
$$\operatorname { Cot } \mathrm { A } / \mathrm { b } + \cot \mathrm { B } / 2 + \cot \mathrm { C } / 2 = \mathrm { A } / 2 \cot \mathrm {~B} / 2 \cot \mathrm { C } / 2$$
(b) Let ABC be a triangle with incentre I and inradius r . Let $\mathrm { D } , \mathrm { E } , \mathrm { F }$ be the feet of the perpendiculars from I to the sides $\mathrm { BC } , \mathrm { CA }$ and AB respectively. If $\mathrm { r } 1 , \mathrm { r } 2$ and B are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that r1/ ( $\mathrm { r } - \mathrm { r } 1$ ) + உ / ( $\mathrm { r } - \mathrm { r } 2$ ) + r3 / ( $\mathrm { r } - \mathrm { r } 3$ ) = ( r 1 B B ) / ( $\mathrm { r } - \mathrm { r } 1$ ) ( $\mathrm { r } - \mathrm { r } 2$ ) ( $\mathrm { r } - \mathrm { r } 3$ )

Let $AB$ and $PQ$ be two vertical poles, 160 m apart from each other. Let $C$ be the middle point of $B$ and $Q$, which are feet of these two poles. Let $\frac{\pi}{8}$ and $\theta$ be the angles of elevation from $C$ to $P$ and $A$, respectively. If the height of pole $PQ$ is twice the height of pole $AB$, then $\tan^2\theta$ is equal to
(1) $\frac{3-2\sqrt{2}}{2}$
(2) $\frac{3+\sqrt{2}}{2}$
(3) $\frac{3-2\sqrt{2}}{4}$
(4) $\frac{3-\sqrt{2}}{4}$

In the regular pentagon ABCDE shown, K and L are the midpoints of line segments AB and DA, respectively.
Given this, what is the measure of angle LKB in degrees?
A) 105
B) 108
C) $\mathbf { 1 2 0 }$
D) 126
E) 135

ABC is an isosceles triangle $\mathrm { D } \in [ \mathrm { BC } ] , \mathrm { B } \in [ \mathrm { AE } ]$ $| \mathrm { AB } | = | \mathrm { BC } |$ $| \mathrm { AC } | = | \mathrm { AD } | = | \mathrm { DE } |$ $\mathrm { m } ( \widehat { \mathrm { ADB } } ) = 111 ^ { \circ }$ $m ( \widehat { B D E } ) = x$
Accordingly, how many degrees is $x$?
A) 15 B) 18 C) 21 D) 24 E) 27

In the figure, the line segments $[OA]$ and $[OD]$ intersect perpendicularly.
Accordingly, the ratio of the area of triangle OAB to the area of triangle OCD in terms of $\alpha$ is which of the following?
A) $\tan \alpha$
B) $\cot \alpha$
C) $\tan^2 \alpha$
D) $\cot^2 \alpha$
E) $\sec^2 \alpha$