Show that the distance between a point on the hyperbola $xy=5$ and a point on the ellipse $x^{2}+6y^{2}=6$ is at least $\frac{9}{7}$.

Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.

Evaluate the following definite integral. \[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]

In $\triangle ABC,\sin A=\frac{3}{5},\cos B=\frac{5}{13}$, then $\cos C=$________.

Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_P$ again at $X$. Similarly, suppose the circle with diameter $\overline{AQ}$ intersects $\omega_Q$ again at $N$, and line $AN$ intersects $\omega_Q$ again at $Y$. Prove that lines $MN$ and $XY$ are parallel. (Here, the points $P$ and $Q$ are [i]isogonal conjugates[/i] with respect to $\triangle ABC$ if the internal angle bisectors of $\angle BAC$, $\angle CBA$, and $\angle ACB$ also bisect the angles $\angle PAQ$, $\angle PBQ$, and $\angle PCQ$, respectively. For example, the orthocenter is the isogonal conjugate of the circumcenter.) [i]Proposed by Sammy Luo[/i]

$S=\sin{64x}+\sin{65x}$ and $C=\cos{64x}+\cos{65x}$ are both rational for some $x$. Prove, that for one of these sums both summands are rational too.

Evaluate $ \int_{ \minus{} 1}^ {a^2} \frac {1}{x^2 \plus{} a^2}\ dx\ (a > 0).$ You may not use $ \tan ^{ \minus{} 1} x$ or Complex Integral here.

Let $a$, $b$, $c$ be positive real numbers. Prove that \[ \biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2 \biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr). \]

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-2.687,0),E); label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$

Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$. If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$.

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies [list] [*]$f(0)=0$ [*]$f(6)=1$ [*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list] Find $f(2)^2+f(10)^2$.

A regular 14-gon with side $a$ is inscribed in a circle of radius one. Prove \[ \frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}. \]

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

Let $n \ge 2$ be an integer. Find all real numbers $a$ such that there exist real numbers $x_1,x_2,\dots,x_n$ satisfying \[x_1(1-x_2)=x_2(1-x_3)=\dots=x_n(1-x_1)=a.\] [i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]

Square $ ABCD$ has sides of length $ 4$, and $ M$ is the midpoint of $ \overline{CD}$. A circle with radius $ 2$ and center $ M$ intersects a circle with raidus $ 4$ and center $ A$ at points $ P$ and $ D$. What is the distance from $ P$ to $ \overline{AD}$? [asy]unitsize(8mm); defaultpen(linewidth(.8pt)); dotfactor=4; draw(Circle((2,0),2)); draw(Circle((0,4),4)); clip(scale(4)*unitsquare); draw(scale(4)*unitsquare); filldraw(Circle((2,0),0.07)); filldraw(Circle((3.2,1.6),0.07)); label("$A$",(0,4),NW); label("$B$",(4,4),NE); label("$C$",(4,0),SE); label("$D$",(0,0),SW); label("$M$",(2,0),S); label("$P$",(3.2,1.6),N);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac {16}{5} \qquad \textbf{(C)}\ \frac {13}{4} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {7}{2}$

A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.

Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits. [img]http://i2.tinypic.com/5x73dza.png[/img]

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+6f(n)=3f(f(n))+4n+2001.\]

In the Cartesian plane, let $C$ be a unit circle with center at origin $O$. For any point $Q$ in the plane distinct from $O$, define $Q'$ to be the intersection of the ray $OQ$ and the circle $C$. Prove that for any $P\in C$ and any $k\in\mathbb{N}$ there exists a lattice point $Q(x,y)$ with $|x|=k$ or $|y|=k$ such that $PQ'<\frac{1}{2k}$.

The circumradius $R$ of a triangle with side lengths $a, b, c$ satisfies $R =\frac{a\sqrt{bc}}{b+c}$. Find the angles of the triangle.

Suppose $(a_1,a_2,a_3,a_4)$ is a 4-term sequence of real numbers satisfying the following two conditions: [list] [*] $a_3=a_2+a_1$ and $a_4=a_3+a_2$; [*] there exist real numbers $a,b,c$ such that \[an^2+bn+c=\cos(a_n)\] for all $n\in\{1,2,3,4\}$. [/list] Compute the maximum possible value of \[\cos(a_1)-\cos(a_4)\] over all such sequences $(a_1,a_2,a_3,a_4)$.