Let the lengths of the three sides of the triangle ABC be $\mathrm { AB } = 6 , \mathrm { BC } = 8$ and $\mathrm { CA } = 4$. Let $\mathrm { O } ^ { \prime }$ be the center of the circle which passes through the two points B and C and is tangent to the straight line AB. Let O be the center of the circle circumscribed about triangle ABC. We are to find the length of the line segment $\mathrm { OO } ^ { \prime }$.
(1) First, we have $\cos \angle \mathrm { ABC } = \frac { \mathbf { A } } { \mathbf { B } }$ and $\sin \angle \mathrm { ABC } = \frac { \sqrt { \mathbf { C D } } } { \mathbf { E } }$.
(2) The radius of the circle circumscribed about triangle ABC is $\frac { \mathbf { F G } \sqrt { \mathbf { H I } } } { \mathbf { J K } }$.
(3) When the intersection point of the straight line $\mathrm { OO } ^ { \prime }$ and the side BC is denoted by D, we have
$$\mathrm { OD } = \frac { \mathbf{N} \sqrt { \mathbf { L M } } } { \mathbf { O P } } \text { and } \mathrm { O } ^ { \prime } \mathrm { D } = \frac { \mathbf { Q R } \sqrt { \mathbf { S T } } } { \mathbf { U V } } .$$
Thus we have $\mathrm { OO } ^ { \prime } = \frac { \mathbf { W } \sqrt { \mathbf { X Y } } } { \mathbf { Z } }$.

[1] In $\triangle \mathrm { ABC }$, $\mathrm { AB } = \sqrt { 3 } - 1$, $\mathrm { BC } = \sqrt { 3 } + 1$, and $\angle \mathrm { ABC } = 60 ^ { \circ }$.
(1) $\mathrm { AC } = \sqrt { \text { A } }$, so the circumradius of $\triangle \mathrm { ABC }$ is $\sqrt { \text { B } }$, and
$$\sin \angle \mathrm { BAC } = \frac { \sqrt { \square } + \sqrt { \square } } { \square }$$
Here, the order of answers for □ C and □ D does not matter.
(2) Let D be a point on side AC such that the area of $\triangle \mathrm { ABD }$ is $\frac { \sqrt { 2 } } { 6 }$. Then
$$\mathrm { AB } \cdot \mathrm { AD } = \frac { \square \sqrt { * } - \square } { \square }$$
Therefore, $\mathrm { AD } = \frac { \square } { \square }$. □ E F
[2] Ski jumping is a competition where athletes compete on both jump distance and the beauty of their aerial posture. Athletes slide down a slope and take off from the edge of the slope into the air. A score $X$ is determined from the jump distance $D$ (in meters), and a score $Y$ is determined from the aerial posture. Consider 58 jumps at a certain competition.
(1) For the score $X$, the score $Y$, and the takeoff velocity $V$ (in km/h), three scatter plots in Figure 1 were obtained.
Choose one from options (0)–(6) below for each of G, H, and I. The order of answers does not matter.
The correct statements that can be read from Figure 1 are □ G, □ H, and □ I. (0) The correlation between $X$ and $V$ is stronger than the correlation between $X$ and $Y$.
(1) There is a positive correlation between $X$ and $Y$.
(2) The jump with maximum $V$ also has maximum $X$.
(3) The jump with maximum $V$ also has maximum $Y$.
(4) The jump with minimum $Y$ does not have minimum $X$.
(5) All jumps with $X \geq 80$ have $V \geq 93$. (6) There is no jump with $Y \geq 55$ and $V \geq 94$.
(2) The score $X$ is calculated from the jump distance $D$ using the following formula:
$$X = 1.80 \times ( D - 125.0 ) + 60.0$$
Choose one from options (0)–(6) below for each of J, K, and L. You may select the same option more than once. – The variance of $X$ is □ J times the variance of $D$. – The covariance of $X$ and $Y$ is □ K times the covariance of $D$ and $Y$. Here, covariance is defined as the average of the products of deviations from the mean for each of the two variables. – The correlation coefficient of $X$ and $Y$ is □ L times the correlation coefficient of $D$ and $Y$. (0) $- 125$
(1) $- 1.80$
(2) $1$
(3) $1.80$
(4) $3.24$
(5) $3.60$ (6) $60.0$
(3) The 58 jumps were performed by 29 athletes, each performing 2 jumps. A histogram for the values of $X + Y$ (sum of scores $X$ and $Y$) for the first jump and a histogram for the values of $X + Y$ for the second jump are either A or B in Figure 2. Also, a box plot for the values of $X + Y$ for the first jump and a box plot for the values of $X + Y$ for the second jump are either a or b in Figure 3. The minimum value of $X + Y$ for the first jump was 108.0.
Choose one from options ⓪–③ in the table below for the blank M.
The correct combination of histogram and box plot for the values of $X + Y$ for the first jump is □ M.

\cline { 2 - 5 } \multicolumn{1}{c|}{}	(0)	(1)	(2)	(3)
Histogram	A	A	B	B
Box plot	a	b	a	b

Choose one from options (0)–(3) below for the blank N.
The correct statement that can be read from Figure 3 is □ N. (0) The interquartile range of $X + Y$ for the first jump is larger than the interquartile range of $X + Y$ for the second jump.
(1) The median of $X + Y$ for the first jump is larger than the median of $X + Y$ for the second jump.
(2) The maximum value of $X + Y$ for the first jump is smaller than the maximum value of $X + Y$ for the second jump.
(3) The minimum value of $X + Y$ for the first jump is smaller than the minimum value of $X + Y$ for the second jump.

The triangle ABC satisfies
$$\mathrm { AB } = 4 , \quad \mathrm { AC } = 3 \quad \text { and } \quad \angle \mathrm { B } = 30 ^ { \circ } .$$
D is the point on side BC such that $\mathrm { AC } = \mathrm { AD }$. Let us consider the circumscribed circle O of triangle ACD.
(1) Since $\sin B = \frac { \mathbf { A } } { \mathbf { B } }$, we have $\sin C = \frac { \mathbf { C } } { \mathbf{D} }$.
Hence the radius of circle O is $\frac { \mathbf { E } } { \mathbf{F} }$.
(2) We have
$$\mathrm { BC } = \mathrm { G } \sqrt { \mathrm { H } } + \sqrt { \mathrm{H} }$$
and
$$\mathrm { BD } = \mathrm { J } \sqrt { \mathrm {~K} } - \sqrt { \mathrm { K } } .$$
Let us denote the intersection of side AB and circle O by E . Then
$$\mathrm { BE } = \frac { \mathbf { M } } { \mathbf{N} } .$$
Hence the relationships between the areas of triangles $\mathrm { BDE } , \mathrm { ADE }$ and ACD are
$$\begin{aligned} & \triangle \mathrm { BDE } : \triangle \mathrm { ADE } = \mathbf { O } : \mathbf { P } , \\ & \triangle \mathrm { BDE } : \triangle \mathrm { ACD } = \mathbf { Q } ( \mathbf{J} \sqrt { \mathbf { K } } - \sqrt { \mathrm { K } } ) : \mathbf { R S } \sqrt { \mathbf { T } } . \end{aligned}$$

Let the quadrangle ABCD be a rhombus where the length of the sides is $\sqrt { 2 }$ and $\angle \mathrm { ABC } = 30 ^ { \circ }$.
(1) We have
$$\mathrm { AC } ^ { 2 } = \mathbf { A } - \mathbf { B } \sqrt { \mathbf { C } } , \quad \mathrm { BD } ^ { 2 } = \mathbf { E } + \mathbf{F} \sqrt{\mathbf{E}} .$$
Now, for any positive numbers $a$ and $b$, we have
$$( \sqrt { a } \pm \sqrt { b } ) ^ { 2 } = a + b \pm 2 \sqrt { a b } \quad \text { (double-sign correspondence). }$$
Using this formula, we obtain
$$\mathrm { AC } = \sqrt { \mathbf { G } } - \mathbf { H } , \quad \mathrm { BD } = \sqrt { \mathbf { I } } + \mathbf{I} . \mathbf { J } .$$
(2) Let us draw four circles, each centered on one vertex of rhombus ABCD, with the following conditions:
The radii of the circles centered on vertices A and C are of length $r$, and those centered on vertices B and D are of length $\sqrt { 2 } - r$.
Circles centered on opposite vertices (A and C, B and D) may touch each other but may not intersect.
Let us denote the area of the region common to rhombus ABCD and these four circles by $S$. We have
$$S = \pi \left( r ^ { 2 } - \frac { \sqrt { \mathbf { K } } } { \mathbf { L } } r \right)$$
where the range of $r$ is
$$\sqrt { \mathbf { O } } - \frac { \sqrt { \mathbf { P } } + \mathbf { Q } } { \mathbf { R } } \leqq r \leqq \frac { \sqrt { \mathbf { S } } - \square \mathbf { T } } { \square \mathbf { U } }$$
Hence $S$ is minimized when $r = \frac { \sqrt { \mathbf { V } } } { \mathbf { W } }$, and the value of $S$ then is $\frac { \mathbf{X} } { \mathbf { Y } } \pi$.

For a quadrilateral ABCD inscribed in a circle of radius 1, let $\mathrm { AB } : \mathrm { AD } = 1 : 2$ and $\angle \mathrm { BAD } = 120 ^ { \circ }$. Also, when the point of intersection of diagonals BD and AC is denoted by E, let $\mathrm { BE } : \mathrm { ED } = 3 : 4$.
We are to find the area of quadrilateral $\mathrm{ABCD}$.
In order to find the area of quadrilateral ABCD, we are to find the area of triangle ABD, denoted by $\triangle \mathrm { ABD }$, and the area of triangle BCD, denoted by $\triangle \mathrm { BCD }$.
First, let us find $\triangle \mathrm { ABD }$. Since
$$\mathrm { BD } = \sqrt { \mathbf { A } } , \quad \mathrm { AB } = \frac { \sqrt { \mathbf { BC } } } { \mathbf { D } } ,$$
we have
$$\triangle \mathrm { ABD } = \frac { \mathbf { E } \sqrt { \mathbf { F } } } { \mathbf { GH } } .$$
Next, let us find $\triangle \mathrm { BCD }$. Since
$$\triangle \mathrm { ABC } : \triangle \mathrm { ACD } = \mathbf{I} : \mathbf { J } ,$$
we see that $\mathrm { BC } : \mathrm { CD } = \mathbf { K } : \mathbf { L }$. (Give the answers using the simplest integer ratios.)
Hence we have $\mathrm { BC } = \frac { \mathbf{M} \sqrt { \mathbf { N } } } { \mathbf{O} }$ and
$$\triangle \mathrm { BCD } = \frac { \mathbf { PQ } \sqrt { \mathbf { R } } } { \mathbf { ST } }$$
Thus, from (1) and (2) we obtain the result that the area of quadrilateral ABCD is $\frac{\mathbf{U}\sqrt{\mathbf{V}}}{\mathbf{WX}}$.

Consider a triangle ABC and its circumscribed circle O, where the lengths of the three sides of the triangle are $$\mathrm{AB} = 2, \quad \mathrm{BC} = 3, \quad \mathrm{CA} = 4.$$ Below, the area of a triangle such as PQR is expressed as $\triangle\mathrm{PQR}$.
(1) We see that $\cos\angle\mathrm{ABC} = \frac{\mathbf{AB}}{\mathbf{C}}$.
(2) Let us take a point D on the circumference of circle O such that it is on the opposite side of the circle from point B with respect to AC and $$\frac{\triangle\mathrm{ABD}}{\triangle\mathrm{BCD}} = \frac{8}{15}.$$ We are to find the lengths of line segments AD and CD.
First, since $$\angle\mathrm{BAD} = \mathbf{DEF}^\circ - \angle\mathrm{BCD},$$ we have $\sin\angle\mathrm{BAD} = \sin\angle\mathrm{BCD}$. Hence from (1) we have $$\frac{\mathrm{AD}}{\mathrm{CD}} = \frac{\mathbf{G}}{\mathbf{H}},$$ so we set $\mathrm{AD} = \mathbf{G}k$ and $\mathrm{CD} = \mathbf{H}k$, where $k$ is a positive number. Furthermore, since $$\angle\mathrm{ADC} = \mathbf{IJK}^\circ - \angle\mathrm{ABC},$$ we have $\cos\angle\mathrm{ADC} = \frac{\mathbf{L}}{\mathbf{L}}$. Hence, we obtain $k = \frac{\mathbf{N}}{\sqrt{\mathbf{OP}}}$, and then $$\mathrm{AD} = \frac{\mathbf{QR}\sqrt{\mathbf{OP}}}{\mathbf{OP}},$$ $$\mathrm{CD} = \frac{\mathbf{ST}\sqrt{\mathbf{OP}}}{\mathbf{OP}}.$$
(3) When we denote the point of intersection of the straight line DA and the straight line CB by E, we have $$\frac{\triangle\mathrm{ABE}}{\triangle\mathrm{CDE}} = \frac{\mathbf{UV}}{\mathbf{WXY}}.$$

In a triangle ABC, let $\angle \mathrm { B } = 45 ^ { \circ }$ and $\angle \mathrm { C } = 75 ^ { \circ }$, and let D be the intersection of the bisector of $\angle \mathrm { A }$ and side BC.
(1) From the law of sines we have
$$\mathrm { AC } = \frac { \sqrt { \mathbf { A } } } { \sqrt { \mathbf { B } } } \mathrm { BC } , \quad \mathrm { AD } = \sqrt { \mathbf { C } } \mathrm { BD } .$$
In particular, from $\angle \mathrm { ADC } = \mathbf { D E } ^ { \circ }$ we see that
$$\mathrm { BD } : \mathrm { BC } = \mathbf { F } : \sqrt { \mathbf { G } }$$
and hence we have
$$\mathrm { AB } : \mathrm { AC } = \mathbf { H } : \left( \sqrt { \mathbf { I } } - \frac { \mathbf { J } }{\mathbf{J} } \right) .$$
(2) Let $\mathrm { O } _ { 1 }$ be the center of the circumscribed circle of triangle ABD, and let $\mathrm { O } _ { 2 }$ be the center of the circumscribed circle of triangle ADC. Let us find the ratio of the areas of triangle ABC and triangle $\mathrm { AO } _ { 1 } \mathrm { O } _ { 2 }$, $\triangle \mathrm { ABC } : \triangle \mathrm { AO } _ { 1 } \mathrm { O } _ { 2 }$.
Since $\angle \mathrm { AO } _ { 1 } \mathrm { D } = \mathbf { K } \mathbf { L } ^ { \circ }$ and $\angle \mathrm { AO } _ { 2 } \mathrm { O } _ { 1 } = \mathbf { M N } ^ { \circ }$, by the same reasoning as (1), we have
$$\mathrm { AC } = \sqrt { \mathbf { O } } \mathrm { AO } _ { 1 } , \quad \mathrm { AO } _ { 2 } = ( \sqrt { \mathbf { P } } - \mathbf { Q } ) \mathrm { AO } _ { 1 } .$$
Hence we obtain
$$\triangle \mathrm { ABC } : \triangle \mathrm { AO } _ { 1 } \mathrm { O } _ { 2 } = \mathbf { R } : ( \mathbf { S } - \sqrt { \mathbf { T } } ) .$$

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Maths \& Computer Science and Computer Science applicants should turn to page 14. [Figure] [Figure]
A triangle $A B C$ has sides $B C , C A$ and $A B$ of sides $a , b$ and $c$ respectively, and angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$ where $0 \leqslant \alpha , \beta , \gamma \leqslant \frac { 1 } { 2 } \pi$.
(i) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.
Deduce the sine rule
$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$
(ii) The points $P , Q$ and $R$ are respectively the feet of the perpeniculars from $A$ to $B C$, $B$ to $C A$, and $C$ to $A B$ as shown.
Prove that
$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$
(iii) For what triangles $A B C$, with angles $\alpha , \beta , \gamma$ as above, does the equation
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$
hold?

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Maths \& Computer Science and Computer Science applicants should turn to page 14. [Figure] [Figure]
A triangle $A B C$ has sides $B C , C A$ and $A B$ of sides $a , b$ and $c$ respectively, and angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$ where $0 \leqslant \alpha , \beta , \gamma \leqslant \frac { 1 } { 2 } \pi$.
(i) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.
Deduce the sine rule
$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$
(ii) The points $P , Q$ and $R$ are respectively the feet of the perpeniculars from $A$ to $B C$, $B$ to $C A$, and $C$ to $A B$ as shown.
Prove that
$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$
(iii) For what triangles $A B C$, with angles $\alpha , \beta , \gamma$ as above, does the equation
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$
hold?

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.

Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 14.
In the diagram below is sketched a semicircle with centre $B$ and radius 1 . Three points $A , C , D$ lie on the semicircle as shown with $\alpha$ denoting angle $C A B$ and $\beta$ denoting angle $D A B$. The triangles $A B C$ and $A B D$ intersect in a triangle $A B X$.
Throughout the question we shall consider the value of $\alpha$ fixed. Assume for now that $0 < \alpha \leqslant \beta \leqslant \pi / 2$. [Figure]
(i) Show that the area of the triangle $A B C$ equals
$$\frac { 1 } { 2 } \sin ( 2 \alpha ) .$$
(ii) Let
$$F = \frac { \text { area of triangle } A B X } { \text { area of triangle } A B C }$$
Without calculation, explain why, for every $k$ in the range $0 \leqslant k \leqslant 1$, there is a unique value of $\beta$ such that $F = k$.
(iii) Find the value of $\beta$ such that $F = 1 / 2$.
(iv) Show that
$$F = \frac { \sin ( 2 \beta ) \sin \alpha } { \sin ( 2 \beta - \alpha ) \sin ( 2 \alpha ) }$$
(v) Suppose now that $0 < \beta < \alpha \leqslant \pi / 2$. Write down, without further calculation, an expression for the area of $A B X$ and hence a formula for $F$.

3. The figure on the right is formed by stacking three right triangles, and $\overline { O D } = 8$. Question: What is the height $\overline { A B }$ of right triangle $O A B$?
(1) 1
(2) $\sqrt { 6 } - \sqrt { 2 }$
(3) $\sqrt { 7 } - 1$
(4) $\sqrt { 3 }$
(5) 2 [Figure]

5. Assume that the distances between towns A, B, and C are all equal to 20 kilometers. Two straight roads intersect at town D, one passing through towns A and B, and the other passing through town C. On an accurately scaled map, the angle between the two roads is measured to be $45^{\circ}$. Then the distance between towns C and D is approximately
(1) 24.5 kilometers
(2) 25 kilometers
(3) 25.5 kilometers
(4) 26 kilometers
(5) 26.5 kilometers

From 6, 8, 10, 12, select any three distinct numbers as the three sides of a triangle, and let $\theta$ be the largest interior angle of this triangle. Among all possible triangles formed, the minimum value of $\cos \theta$ is (Express as a fraction in lowest terms)

In $\triangle A B C$, it is known that $\overline { A B } = 4$ and $\overline { A C } = 6$, which is insufficient to determine the shape and size of $\triangle A B C$. However, knowing certain additional conditions (for example, knowing the length of $\overline { B C }$) would uniquely determine the shape and size of $\triangle A B C$. Select the correct options.
(1) If we additionally know the value of $\cos A$, then $\triangle A B C$ can be uniquely determined
(2) If we additionally know the value of $\cos B$, then $\triangle A B C$ can be uniquely determined
(3) If we additionally know the value of $\cos C$, then $\triangle A B C$ can be uniquely determined
(4) If we additionally know the area of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined
(5) If we additionally know the circumradius of $\triangle A B C$, then $\triangle A B C$ can be uniquely determined

On the coordinate plane, the three vertices of $\triangle A B C$ have coordinates $A ( 0,2 ) , B ( 1,0 ) , C ( 4,1 )$ respectively. Select the correct options.
(1) Among the three sides of $\triangle A B C$, $\overline { A C }$ is the longest
(2) $\sin A < \sin C$
(3) $\triangle A B C$ is an acute triangle
(4) $\sin B = \frac { 7 \sqrt { 2 } } { 10 }$
(5) The circumradius of $\triangle A B C$ is less than 2

As shown in the diagram on the right, there is a $\triangle ABC$. It is known that the altitude $\overline{AD} = 12$ on side $\overline{BC}$, and $\tan \angle B = \frac{3}{2}$, $\tan \angle C = \frac{2}{3}$. What is the length of $\overline{BC}$?
(1) 20
(2) 21
(3) 24
(4) 25
(5) 26

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 circle, 12 equally spaced points are marked and numbered 1 to 12 in clockwise order. Among all triangles formed by choosing any 3 of these 12 points as vertices, the number of triangles whose three interior angles, arranged from smallest to largest, form an arithmetic sequence is (17-1)(17-2).

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 $c$

In $\triangle A B C$, $\overline { A B } = 6 , \overline { A C } = 5 , \overline { B C } = 4$. Let $D$ be the midpoint of $\overline { A B }$, and $P$ be the intersection of the angle bisector of $\angle A B C$ and $\overline { C D }$, as shown in the figure. Select the correct options.
(1) $\overline { C P } = \frac { 3 } { 7 } \overline { C D }$
(2) $\overrightarrow { A P } = \frac { 3 } { 7 } \overrightarrow { A B } + \frac { 2 } { 7 } \overrightarrow { A C }$
(3) $\cos \angle B A C = \frac { 3 } { 4 }$
(4) The area of $\triangle A C P$ is $\frac { 15 } { 14 } \sqrt { 7 }$
(5) (Dot product) $\overrightarrow { A P } \cdot \overrightarrow { A C } = \frac { 120 } { 7 }$

In $\triangle A B C$, $\overline { A B } = \overline { B C } = 3$ and $\cos \angle A B C = - \frac { 1 } { 8 }$. On the circumcircle of $\triangle A B C$ there is a point $D$ satisfying $\overline { B D } = 4$ and $\overline { A D } \leq \overline { C D }$. Then $\overline { C D } = $ (17-1) $+$ $\sqrt{\text{(17-2)}}$. (Express as a simplified radical form.)

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 $P R Q < 180$

In the triangle $P Q R , P R = 2 , Q R = p$ and $\angle R P Q = 30 ^ { \circ }$.
What is the set of all the values of $p$ for which this information uniquely determines the length of $P Q$ ?

A triangle $A B C$ is to be drawn with $A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}$ and the angle at $A$ equal to $\theta$, where $\theta$ is a certain specified angle.
Of the two possible triangles that could be drawn, the larger triangle has three times the area of the smaller one.
What is the value of $\cos \theta$ ?
A $\frac { 5 } { 7 }$
B $\frac { 151 } { 200 }$
C $\frac { 2 \sqrt { 2 } } { 5 }$
D $\frac { \sqrt { 17 } } { 5 }$
E $\quad \frac { \sqrt { 51 } } { 8 }$
F $\frac { \sqrt { 34 } } { 8 }$

A student chooses two distinct real numbers $x$ and $y$ with $0 < x < y < 1$. The student then attempts to draw a triangle $A B C$ with:
$$\begin{aligned} A B & = 1 \\ \sin A & = x \\ \sin B & = y \end{aligned}$$
Which of the following statements is/are correct?
I For some choice of $x$ and $y$, there is exactly one triangle the student could draw.
II For some choice of $x$ and $y$, there are exactly two different triangles the student could draw.
III For some choice of $x$ and $y$, there are exactly three different triangles the student could draw. (Note that congruent triangles are considered to be the same.)
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III