[i](8 pts)[/i] Let $n$ be a positive integer, and let $a, b, c$ be real numbers. (a) [i](2 pts)[/i] Given that $a\cos x+b\cos 2x +c\cos 3x \geq -1$ for all reals $x$, find, with proof, the maximum possible value of $a+b+c$. (b) [i](6 pts)[/i] Let $f$ be a degree $n$ polynomial with real coefficients. Suppose that $|f(z)| \leq \left|f(z)+\frac{2}{z}\right|$ for all complex $z$ lying on the unit circle. Find, with proof, the maximum possible value of $f(1)$.

A triangle $KLM$ is given in the plane together with a point $A$ lying on the half-line opposite to $KL$. Construct a rectangle $ABCD$ whose vertices $B, C$ and $D$ lie on the lines $KM, KL$ and $LM$, respectively. (We allow the rectangle to be a square.)

Which of the following polynomials with integer coefficients has $\sin\left(10^{\circ}\right)$ as a root? $\text{(A) }4x^3-4x+1\qquad\text{(B) }6x^3-8x^2+1\qquad\text{(C) }4x^3+4x-1\qquad\text{(D) }8x^3+6x-1\qquad\text{(E) }8x^3-6x+1$

Two concentric circles have radii $1$ and $4$. Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$. The radius of the six congruent circles can be written $\textstyle\frac{k+\sqrt m}n$, where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$. [asy] size(150); defaultpen(linewidth(0.8)); real r = (sqrt(133)-9)/2; draw(circle(origin,1)^^circle(origin,4)); for(int i=0;i<=2;i=i+1) { filldraw(circle(dir(90 + i*120)*(4-r),r),gray); } for(int j=0;j<=2;j=j+1) { filldraw(circle(dir(30+j*120)*(1+r),r),darkgray); } [/asy]

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$, let $P$ be a point on $BC$, and let $Q$ be the point (other than $A$) where the line through $A$ and $P$ intersects the circumcircle of $ABC$. Prove that \[PQ\le \frac{25}{4\sqrt{6}}.\]

Solve the equation $ \sin^6x \plus{} \cos^6x \equal{} \frac{1}{4}$.

Prove that if the relationship between the sides and opposite angles $ A $ and $ B $ of the triangle $ ABC $ is $$ (a^2 + b^2) \sin (A - B) = (a^2 - b^2) \sin (A + B)$$ then such a triangle is right-angled or isosceles.

Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$

In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$

Let $C$ be the curve expressed by $x=\sin t,\ y=\sin 2t\ \left(0\leq t\leq \frac{\pi}{2}\right).$ (1) Express $y$ in terms of $x$. (2) Find the area of the figure $D$ enclosed by the $x$-axis and $C$. (3) Find the volume of the solid generated by a rotation of $D$ about the $y$-axis.

Let $a$, $b$, $c$ be the three sides of a triangle, and let $\alpha$, $\beta$, $\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find \[ \frac{\cot \gamma}{\cot \alpha+\cot \beta}. \]

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

Prove no three lattice points in the plane form an equilateral triangle.

Let $a$ be parameter such that $0<a<2\pi$. For $0<x<2\pi$, find the extremum of $F(x)=\int_{x}^{x+a} \sqrt{1-\cos \theta}\ d\theta$.

The incircle of a triangle $ABC$ touches the sides $BC$, $CA$, $AB$ at the points $D$, $E$, $F$ respectively. Let the line $AD$ intersect this incircle of triangle $ABC$ at a point $X$ (apart from $D$). Assume that this point $X$ is the midpoint of the segment $AD$, this means, $AX = XD$. Let the line $BX$ meet the incircle of triangle $ABC$ at a point $Y$ (apart from $X$), and let the line $CX$ meet the incircle of triangle $ABC$ at a point $Z$ (apart from $X$). Show that $EY = FZ$.

How many triangles have area $ 10$ and vertices at $ (\minus{}5,0)$, $ (5,0)$, and $ (5\cos \theta, 5\sin \theta)$ for some angle $ \theta$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 8$

Let $ABC$ be an isosceles triangle with $AB=AC$. Consider a variable point $P$ on the extension of the segment $BC$ beyound $B$ (in other words, $P$ lies on the line $BC$ such that the point $B$ lies inside the segment $PC$). Let $r_{1}$ be the radius of the incircle of the triangle $APB$, and let $r_{2}$ be the radius of the $P$-excircle of the triangle $APC$. Prove that the sum $r_{1}+r_{2}$ of these two radii remains constant when the point $P$ varies. [i]Remark.[/i] The $P$-excircle of the triangle $APC$ is defined as the circle which touches the side $AC$ and the [i]extensions[/i] of the sides $AP$ and $CP$.

A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length $7$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle APD = 60^{\circ}$ and $BP = 8$, then $r^{2} =$ $ \textbf{(A)}\ 70\qquad\textbf{(B)}\ 71\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 73\qquad\textbf{(E)}\ 74 $

Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$

Let $ABCD$ be a convex quadrilateral with $\angle ABC = 120^{\circ}$ and $\angle BCD = 90^{\circ}$, and let $M$ and $N$ denote the midpoints of $\overline{BC}$ and $\overline{CD}$. Suppose there exists a point $P$ on the circumcircle of $\triangle CMN$ such that ray $MP$ bisects $\overline{AD}$ and ray $NP$ bisects $\overline{AB}$. If $AB + BC = 444$, $CD = 256$ and $BC = \frac mn$ for some relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Michael Ren[/i]

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy] size(130); pair A, B, C, D, E, F, G, H, I, J, K, L; A = dir(120); B = dir(60); C = dir(0); D = dir(-60); E = dir(-120); F = dir(180); draw(A--B--C--D--E--F--cycle); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); G = midpoint(A--B); H = midpoint(B--C); I = midpoint(C--D); J = midpoint(D--E); K = midpoint(E--F); L = midpoint(F--A); draw(G--H--I--J--K--L--cycle); label("$A$", A, dir(120)); label("$B$", B, dir(60)); label("$C$", C, dir(0)); label("$D$", D, dir(-60)); label("$E$", E, dir(-120)); label("$F$", F, dir(180)); [/asy] $\textbf{(A)}\ \displaystyle \frac{1}{2} \qquad \textbf{(B)}\ \displaystyle \frac{\sqrt 3}{3} \qquad \textbf{(C)}\ \displaystyle \frac{2}{3} \qquad \textbf{(D)}\ \displaystyle \frac{3}{4} \qquad \textbf{(E)}\ \displaystyle \frac{\sqrt 3}{2}$

Given $ x,y,z\in (0,1)$ satisfying that $ \sqrt{\frac{1 \minus{} x}{yz}} \plus{} \sqrt{\frac{1 \minus{} y}{xz}} \plus{} \sqrt{\frac{1 \minus{} z}{xy}} \equal{} 2$. Find the maximum value of $ xyz$.