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Q.1 In triangle ABC

$3 \sin A + 4 \cos B =6$ and $4 \sin B + 3 \cos A = 1.$

Find the measure of angle C.

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Selected	Possible Answer

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Q.2 In triangle ABC, $\frac{q}{b} = 2 + \sqrt{3}$ and $\angle C = 60^0$ .

Find the measure of angles A and B.

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Selected	Possible Answer
	and
	and
	and
	and

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Q.3 Evaluate

$\cos a \cos 2a \cos 3a\cdots \cos 999a,$

where $a = \frac{2\pi}{1999.}$

Possible Answers

Selected	Possible Answer
	$P = \frac{1}{2^{999}}$
	$P = \frac{1}{2^{888}}$
	$P = \frac{1}{3^{888}}$
	$P = \frac{1}{3^{999}}$

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Q.4 Determine the minimum value of

$\frac{sec^4 \alpha}{tan^2 \beta} + \frac{sec^4 \beta}{tan^2 \alpha}$

Overall $\alpha, \beta \not= \frac{k\pi}{2}$ , where k is in Z.

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Selected	Possible Answer

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Q.5 Find all pairs (x, y) of real numbers with $0 < x < \frac{\pi}{2}$ such that

$\frac{(\sin x)^{2y}}{(\cos x)^{\frac{y^2}{2}}} + \frac{(\cos x^){2y}}{(\sin x^){\frac{y^2 }{2}}} = \sin 2x.$

Possible Answers

Selected	Possible Answer
	$\big(\frac{\pi}{1}, 2 \big).$
	$\big(\frac{\pi}{4}, 2 \big).$
	$\big(\frac{\pi}{2}, 2 \big).$
	$\big(\frac{\pi}{3}, 2 \big).$

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Q.6 [USAMO 2002 proposal, by Cecil Rousseau] Find the maximum value of

S = (1 - x_1)(1 - y_1) + (1 - x_2)(1 - y_2)

if x_2^1 + x_2^2 = y_2^1 + y_2^2 = c^2

Possible Answers

Selected	Possible Answer
	$\frac{-b\sqrt{4}}{2}$
	$\frac{-c\sqrt{1}}{2}$
	$\frac{-c\sqrt{2}}{2}$
	$\frac{-a\sqrt{3}}{2}$

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Q.7 [China 1997] Let x, y, z be real numbers with $x\quad \underline >\quad y\quad \underline >\quad z\quad \underline >\quad \frac{\pi}{12}$ such that $x + y + z = \frac{\pi}{2}$ . Find the maximum and the minimum values of the product $\cos x \sin y \cos z.$

Possible Answers

Selected	Possible Answer
	$x = y = \frac{2\pi}{24}$ and $z = \frac{\pi}{10}$
	$x = y = \frac{5\pi}{20}$ and $z = \frac{\pi}{6}$
	$x = y = \frac{5\pi}{24}$ and $z = \frac{\pi}{12}$
	$x = y = \frac{3\pi}{24}$ and $z = \frac{\pi}{12}$

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Q.8 Find the sum of all x in the interval $[0, 2\pi]$ such that

$3 \cot^2 x + 8 \cot x + 3 = 0.$

Possible Answers

Selected	Possible Answer
	$2\pi$
	$3\pi$
	$4\pi$
	$5\pi$

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Q.9 [AMC12P 2002] Given that

$(1+\tan 1^0)(1+\tan 2^0)\cdots (1+\tan 45^0)=2^n,$

find n.

Possible Answers

Selected	Possible Answer
	23
	21
	25
	22

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Q.10 [AIME 2003] Let A=(0,0) and B=(b,2) be points in the coordinate plane. Let ABCDEF be a convex equilateral hexagon such that $\angle FAB=120^0,AB||DE,BC||EF,$ and CD||FA, and the y coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10 \}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.

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