Let $f: \mathbb{R}\to\mathbb{R}$ is a function such that $f( \cot x ) = \cos 2x+\sin 2x$ for all $0 < x < \pi$. Define $g(x) = f(x) f(1-x)$ for $-1 \leq x \leq 1$. Find the maximum and minimum values of $g$ on the closed interval $[-1, 1].$

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$. Suppose that the radius of the circle is 5, that $BC = 6$, and that $AD$ is bisected by $BC$. Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$. It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $m/n$ in lowest terms, what is the product $mn$? [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(200), D=dir(95), M=midpoint(A--D), C=dir(30), BB=C+2*dir(C--M), B=intersectionpoint(M--BB, Circle(origin, 1)); draw(Circle(origin, 1)^^A--D^^B--C); real r=0.05; pair M1=midpoint(M--D), M2=midpoint(M--A); draw((M1+0.1*dir(90)*dir(A--D))--(M1+0.1*dir(-90)*dir(A--D))); draw((M2+0.1*dir(90)*dir(A--D))--(M2+0.1*dir(-90)*dir(A--D))); pair point=origin; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D));[/asy]

Regular hexagon $ ABCDEF$ has vertices $ A$ and $ C$ at $ (0,0)$ and $ (7,1)$, respectively. What is its area? $ \textbf{(A) } 20\sqrt {3} \qquad \textbf{(B) } 22\sqrt {3} \qquad \textbf{(C) } 25\sqrt {3} \qquad \textbf{(D) } 27\sqrt {3} \qquad \textbf{(E) } 50$

For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions: [b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} > 1$. [b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta}, \frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 - x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} + \lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta} \rbrack^{2} \leq a$.

Squares $CAKL$ and $CBMN$ are constructed on the sides of acute-angled triangle $ABC$, outside of the triangle. Line $CN$ intersects line segment $AK$ at $X$, while line $CL$ intersects line segment $BM$ at $Y$. Point $P$, lying inside triangle $ABC$, is an intersection of the circumcircles of triangles $KXN$ and $LYM$. Point $S$ is the midpoint of $AB$. Prove that angle $\angle ACS=\angle BCP$.

Evaluate \[\int_0^{\pi} xp^x\cos qx\ dx,\ \int_0^{\pi} xp^x\sin qx\ dx\ (p>0,\ p\neq 1,\ q\in{\mathbb{N^{+}}})\] Own

Points $A$ and $B$ lie on circle $\omega$. Point $P$ lies on the extension of segment $AB$ past $B$. Line $\ell$ passes through $P$ and is tangent to $\omega$. The tangents to $\omega$ at points $A$ and $B$ intersect $\ell$ at points $D$ and $C$ respectively. Given that $AB=7$, $BC=2$, and $AD=3$, compute $BP$.

Let $ a$, $ b$, $ c$ be positive real numbers for which $ a \plus{} b \plus{} c \equal{} 1$. Prove that \[ {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}} \leq {3 \over 2}.\]

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$

Two beads, each of mass $m$, are free to slide on a rigid, vertical hoop of mass $m_h$. The beads are threaded on the hoop so that they cannot fall off of the hoop. They are released with negligible velocity at the top of the hoop and slide down to the bottom in opposite directions. The hoop remains vertical at all times. What is the maximum value of the ratio $m/m_h$ such that the hoop always remains in contact with the ground? Neglect friction. [asy] pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,0)--(3,0)); draw(circle((1.5,1),1)); filldraw(circle((1.4,1.99499),0.1), gray(.3)); filldraw(circle((1.6,1.99499),0.1), gray(.3)); [/asy]

Let $ H$ be the piont of midpoint of the cord $ PQ$ that is on the circle centered the origin $ O$ with radius $ 1.$ Suppose the length of the cord $ PQ$ is $ 2\sin \frac {t}{2}$ for the angle $ t\ (0\leq t\leq \pi)$ that is formed by half-ray $ OH$ and the positive direction of the $ x$ axis. Answer the following questions. (1) Express the coordiante of $ H$ in terms of $ t$. (2) When $ t$ moves in the range of $ 0\leq t\leq \pi$, find the minimum value of $ x$ coordinate of $ H$. (3) When $ t$ moves in the range of $ 0\leq t\leq \frac {\pi}{2}$, find the area $ S$ of the region bounded by the curve drawn by the point $ H$ and the $ x$ axis and the $ y$ axis.

The diagram below shows a regular hexagon with an inscribed square where two sides of the square are parallel to two sides of the hexagon. There are positive integers $m$, $n$, and $p$ such that the ratio of the area of the hexagon to the area of the square can be written as $\tfrac{m+\sqrt{n}}{p}$ where $m$ and $p$ are relatively prime. Find $m + n + p$. [asy] import graph; size(4cm); pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((0,1)--(1,1)--(1.5,1.87)--(1,2.73)--(0,2.73)--(-0.5,1.87)--cycle); filldraw((1.13,2.5)--(-0.13,2.5)--(-0.13,1.23)--(1.13,1.23)--cycle,grey); draw((0,1)--(1,1)); draw((1,1)--(1.5,1.87)); draw((1.5,1.87)--(1,2.73)); draw((1,2.73)--(0,2.73)); draw((0,2.73)--(-0.5,1.87)); draw((-0.5,1.87)--(0,1)); draw((1.13,2.5)--(-0.13,2.5)); draw((-0.13,2.5)--(-0.13,1.23)); draw((-0.13,1.23)--(1.13,1.23)); draw((1.13,1.23)--(1.13,2.5)); [/asy]

Evaluate \[\int_0^{\pi} \frac{x^2\cos ^ 2 x-x\sin x-\cos x-1}{(1+x\sin x)^2}dx\]

Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system \begin{align*} x\cos\beta+\frac1z\cos\alpha &=1, \\ y\cos\gamma+\frac1x\cos\beta &=1, \\ z\cos\alpha+\frac1y\cos\gamma &=1. \end{align*}

Find the minimum value of \[|\sin{x} + \cos{x} + \tan{x} + \cot{x} + \sec{x} + \csc{x}|\] for real numbers $x$.

Find all $t\in \mathbb R$, such that $\int_{0}^{\frac{\pi}{2}}\mid \sin x+t\cos x\mid dx=1$ .

ABC is an isosceles triangle with AB=AC. Point D lies on side BC. Point F is inside $\triangle$ABC and lies on the circumcircle of triangle ADC. The circumcircle of triangle BDF intersects side AB at point E. Prove that $CD\cdot EF+DF\cdot AE=BD\cdot AF$.

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

Given a triangle with $120^\circ$. Let $x,\ y,\ z$ be the side lengths of the triangle such that $x<y<z$. (1) Find all triplets $(x,\ y,\ z)$ of positive integers $x,\ y,\ z$ such that $x+y-z=2$. (2) Find all triplets $(x,\ y,\ z)$ of positive integers $x,\ y,\ z$ such that $x+y-z=3$. (3) Let $a,\ b$ be non-negative integers. Express the number of $(x,\ y,\ z)$ such that $x+y-z=2^a3^b$ in terms of $a,\ b$. 2012 Hitotsubashi University entrance exam, problem 1

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

Circles $A$, $B$, and $C$ are externally tangent to each other and internally tangent to circle $D$. Circles $B$ and $C$ are congruent. Circle $A$ has radius 1 and passes through the center of $D$. What is the radius of circle $B$? [asy] size(200); defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(Circle(origin, 2)); draw(Circle((-1,0), 1)); draw(Circle((6/9, 8/9), 8/9)); draw(Circle((6/9, -8/9), 8/9)); label("$A$", (-1.2, -0.2), NE); label("$B$", (6/9, 7/9), N); label("$C$", (6/9, -7/9), S); label("$D$", 2*dir(110), dir(110));[/asy] $ \textbf{(A)}\; \frac23\qquad \textbf{(B)}\; \frac{\sqrt{3}}2\qquad \textbf{(C)}\; \frac78\qquad \textbf{(D)}\; \frac89\qquad \textbf{(E)}\; \frac{1+\sqrt3}3 $

Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.

Prove that every positive $x,y$ and real $a$ satisfy inequality $x^{\sin ^2a} y^{\cos^2a} < x + y$.