For a constant $a$, denote $C(a)$ the part $x\geq 1$ of the curve $y=\sqrt{x^2-1}+\frac{a}{x}$. (1) Find the maximum value $a_0$ of $a$ such that $C(a)$ is contained to lower part of $y=x$, or $y<x$. (2) For $0<\theta <\frac{\pi}{2}$, find the volume $V(\theta)$ of the solid $V$ obtained by revoloving the figure bounded by $C(a_0)$ and three lines $y=x,\ x=1,\ x=\frac{1}{\cos \theta}$ about the $x$-axis. (3) Find $\lim_{\theta \rightarrow \frac{\pi}{2}-0} V(\theta)$. 1992 Tokyo University entrance exam/Science, 2nd exam

Prove that for every positive integer $n,$ the set $\{2,3,4,\ldots,3n+1\}$ can be partitioned into $n$ triples in such a way that the numbers from each triple are the lengths of the sides of some obtuse triangle. [i]Proposed by Canada[/i]

Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$. [i]Proposed by F. Bakharev[/i]

For a real number $x$, let $f(x)=\int_0^{\frac{\pi}{2}} |\cos t-x\sin 2t|\ dt$. (1) Find the minimum value of $f(x)$. (2) Evaluate $\int_0^1 f(x)\ dx$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 2[/i]

A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer? [asy]size(12cm); pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75); pair B = (-1,-1); pair C = (-1,-7); pair A = (-13,-7); path inclined_plane = A--B--C--cycle; draw(inclined_plane, p); real r = 1; // for marking angles draw(arc(A, r, 0, degrees(B-A))); // mark angle label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle real h = 1.2; // height of box real w = 1.9; // width of box path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box // box on slope with label picture box_on_slope; filldraw(box_on_slope, box, light_grey, black); label(box_on_slope, "$m$", (w/2,h/2)); pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB pair T1 = dir(125); // point of tangency with pulley pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box draw(T1--X1); // string add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope); // picture for the hanging box picture hanging_box; filldraw(hanging_box, box, light_grey, black); label(hanging_box, "$M$", (w/2,h/2)); pair T2 = (1,0); pair X2 = (1,-3); draw(T2--X2); // string add(shift(X2-(w/2,h)) * hanging_box); // Draws the actual pulley filldraw(unitcircle, grey, p); // outer boundary of pulley wheel filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]

Suppose $ H$ and $ O$ are the orthocenter and the circumcenter of acute triangle $ ABC$; $ AA_1$, $ BB_1$ and $ CC_1$ are the altitudes of the triangle. Point $ C_2$ is the reflection of $ C$ in $ A_1B_1$. Prove that $ H$, $ O$, $ C_1$ and $ C_2$ are concyclic.

Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$

Consider positive integers $m <n,p < q$ such that $(m, n) = 1, (p, q) = 1$ and satisfy the condition that if $\frac{m}{n}= tan\alpha$ and $\frac{p}{q} = tan\beta$, then $\alpha + \beta = 45^o$. i) Given $m, n$, find $p, q$. ii) Given $n, q$, find $m, p$. ii) Given $m, q$, find $n, p$.

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$

Points $E$, $F$, $G$, and $H$ lie on sides $AB$, $BC$, $CD$, and $DA$ of a convex quadrilateral $ABCD$ such that $\frac{AE}{EB} \cdot \frac{BF}{FC} \cdot \frac{CG}{GD} \cdot \frac{DH}{HA}=1$. Points $A$, $B$, $C$, and $D$ lie on sides $H_1E_1$, $E_1F_1$, $F_1G_1$, and $G_1H_1$ of a convex quadrilateral $E_1F_1G_1H_1$ such that $E_1F_1 \parallel EF$, $F_1G_1 \parallel FG$, $G_1H_1 \parallel GH$, and $H_1E_1 \parallel HE$. Given that $\frac{E_1A}{AH_1}=a$, express $\frac{F_1C}{CG_1}$ in terms of $a$. 

Let $x,y,z$ be real numbers. Find the minimum value of the sum \begin{align*}|\cos(x)|+|\cos(y)|+|\cos(z)|+|\cos(x-y)|+|\cos(y-z)|+|\cos(z-x)|.\end{align*}

Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?

Compute $\sum_{k=0}^{\infty}\int_{0}^{\frac{\pi}{3}}\sin^{2k} x \, dx$.

Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that \[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.

We consider a cube with sides of length 1. Prove that a tetrahedron with vertices in the set of the vertices of the cube has the volume $\dfrac 16$ if and only if 3 of the vertices of the tetrahedron are vertices on the same face of the cube. [i]Dinu Serbanescu[/i]

Triangle $ ABC$ is inscribed in circle $ \omega$. The tangent lines to $ \omega$ at $ B$ and $ C$ meet at $ T$. Point $ S$ lies on ray $ BC$ such that $ AS \perp AT$. Points $ B_1$ and $ C_1$ lie on ray $ ST$ (with $ C_1$ in between $ B_1$ and $ S$) such that $ B_1T \equal{} BT \equal{} C_1T$. Prove that triangles $ ABC$ and $ AB_1C_1$ are similar to each other.

Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$, prove that $AC \perp BD$.

The three distinct points$ B, C, D$ are collinear with C between B and D. Another point A not on the line BD is such that $|AB| = |AC| = |CD|.$ Prove that ∠$BAC = 36$ if and only if $1/|CD|-1/|BD|=1/(|CD| + |BD|)$ .

Let $O$ be the center of a circle of radius $26$, and let $A$, $B$ be two distinct points on the circle, with $M$ being the midpoint of $AB$. Consider point $C$ for which $CO=34$ and $\angle COM=15^\circ$. Let $N$ be the midpoint of $CO$. Suppose that $\angle ACB=90^\circ$. Find $MN$.

Evaluate $ \int_0^{\frac{\pi}{2}} \frac{\cos \theta \minus{}\sin \theta}{(1\plus{}\cos \theta)(1\plus{}\sin \theta)}\ d\theta$

Find the function $f(x)$ such that $f(x)=\cos (2mx)+\int_{0}^{\pi}f(t)|\cos t|\ dt$ for positive inetger $m.$

$ C$ and $ D$ are points on a semicircle. The tangent at $ C$ meets the extended diameter of the semicircle at $ B$, and the tangent at $ D$ meets it at $ A$, so that $ A$ and $ B$ are on opposite sides of the center. The lines $ AC$ and $ BD$ meet at $ E$. $ F$ is the foot of the perpendicular from $ E$ to $ AB$. Show that $ EF$ bisects angle $ CFD$

Given a plane $P$ in space. For a figure $A$, call orthogonal projection the whole of points of intersection between the perpendicular drawn from each point in $A$ and $P$. Answer the following questions. (1) Let a plane $Q$ intersects with the plane $P$ by angle $\theta\ \left(0<\theta <\frac{\pi}{2}\right)$ between the planes, that is to say,　the angles between two lines, is $\theta$, which can be generated by cuttng $P,\ Q$ by a plane which is perpendicular to the line of intersection of $P$ and $Q$.　Find the maximum and minimum length of the orthogonal projection of the line segment in length 1 on $Q$ on to $P$.. (2) Consider $Q$ in (1). Find the area of the orthogonal projection of a equilateral triangle on $Q$ with side length 1 onto $P$. (3) What's the shape of the orthogonal projection $T'$ of a regular tetrahedron $T$ with side length 1 on to $P'$, then find the max area of $T'$.

In triangle $ABC$, angle $A$ is twice as large as angle $B$. $AB = 3$ and $AC = 2$. Calculate $BC$.