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

Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?

Three semicircles of radius $ 1$ are constructed on diameter $ AB$ of a semicircle of radius $ 2$. The centers of the small semicircles divide $ \overline{AB}$ into four line segments of equal length, as shown. What is the area of the shaded region that lies within the large semicircle but outside the smaller semicircles? [asy]import graph; unitsize(14mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dashed=linetype("4 4"); dotfactor=3; pair A=(-2,0), B=(2,0); fill(Arc((0,0),2,0,180)--cycle,mediumgray); fill(Arc((-1,0),1,0,180)--cycle,white); fill(Arc((0,0),1,0,180)--cycle,white); fill(Arc((1,0),1,0,180)--cycle,white); draw(Arc((-1,0),1,60,180)); draw(Arc((0,0),1,0,60),dashed); draw(Arc((0,0),1,60,120)); draw(Arc((0,0),1,120,180),dashed); draw(Arc((1,0),1,0,120)); draw(Arc((0,0),2,0,180)--cycle); dot((0,0)); dot((-1,0)); dot((1,0)); draw((-2,-0.1)--(-2,-0.3),gray); draw((-1,-0.1)--(-1,-0.3),gray); draw((1,-0.1)--(1,-0.3),gray); draw((2,-0.1)--(2,-0.3),gray); label("$A$",A,W); label("$B$",B,E); label("1",(-1.5,-0.1),S); label("2",(0,-0.1),S); label("1",(1.5,-0.1),S);[/asy]$ \textbf{(A)}\ \pi\minus{}\sqrt3 \qquad \textbf{(B)}\ \pi\minus{}\sqrt2 \qquad \textbf{(C)}\ \frac{\pi\plus{}\sqrt2}{2} \qquad \textbf{(D)}\ \frac{\pi\plus{}\sqrt3}{2}$ $ \textbf{(E)}\ \frac{7}{6}\pi\minus{}\frac{\sqrt3}{2}$

Sides $AB,~ BC,$ and $CD$ of (simple*) quadrilateral $ABCD$ have lengths $4,~ 5,$ and $20$, respectively. If vertex angles $B$ and $C$ are obtuse and $\sin C = - \cos B =\frac{3}{5} $, then side $AD$ has length $\textbf{(A) }24\qquad\textbf{(B) }24.5\qquad\textbf{(C) }24.6\qquad\textbf{(D) }24.8\qquad\textbf{(E) }25$ [size=70]*A polygon is called “simple” if it is not self intersecting.[/size]

A circle with center O inscribed in a convex quadrilateral ABCD is tangent to the lines AB, BC, CD, DA at points K, L, M, N respectively. Assume that the lines KL and MN are not parallel and intersect at the point S. Prove that BD is perpendicular OS. I think it is very good and beautiful problem. I solved it without help. I'm wondering is it a well known theorem? Also I'm interested who is the creator of this problem? I'll be glad to see simple solution of this problem.

A rectangular box has width $12$ inches, length $16$ inches, and height $\tfrac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$.

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

Let $ABCD$ by a cyclic quadrilateral with circumcenter $O$. Let $P$ be the point of intersection of the diagonals $AC$ and $BD$, and $K, L, M, N$ the circumcenters of triangles $AOP, BOP$, $COP, DOP$, respectively. Prove that $KL = MN$.

In a right triangle, ratio of the hypotenuse over perimeter of the triangle determines an interval on real numbers. Find the midpoint of this interval? $\textbf{(A)}\ \frac{2\sqrt{2} \plus{}1}{4} \qquad\textbf{(B)}\ \frac{\sqrt{2} \plus{}1}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{2} \minus{}1}{4} \\ \qquad\textbf{(D)}\ \sqrt{2} \minus{}1 \qquad\textbf{(E)}\ \frac{\sqrt{2} \minus{}1}{2}$

Let $ ABC$ be an arbitrary acute angled triangle. For any point $ P$ lying within the triangle, let $ D$, $ E$, $ F$ denote the feet of the perpendiculars from $ P$ onto the sides $ AB$, $ BC$, $ CA$ respectively. Determine the set of all possible positions of the point $ P$ for which the triangle $ DEF$ is isosceles. For which position of $ P$ will the triangle $ DEF$ become equilateral?

Let $ABC$ be an acute triangle with $AB \neq AC$. Let $D$ be the foot of the altitude from $A$ and $\omega$ the circumcircle of the triangle. Let $\omega_1$ be the circle tangent to $AD$, $BD$ and $\omega$. Let $\omega_2$ be the circle tangent to $AD$, $CD$ and $\omega$. Let $\ell$ be the interior common tangent to both $\omega_1$ and $\omega_2$, different from $AD$. Prove that $\ell$ passes through the midpoint of $BC$ if and only if $2BC = AB + AC$.

Given a solid $R$ contained in a semi cylinder with the hight $1$ which has a semicircle with radius $1$ as the base. The cross section at the hight $x\ (0\leq x\leq 1)$ is the form combined with two right-angled triangles as attached figure as below. Answer the following questions. (1) Find the cross-sectional area $S(x)$ at the hight $x$. (2) Find the volume of $R$. If necessary, when you integrate, set $x=\sin t.$

Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

In $ xyz$ space, consider two points $ P(1,\ 0,\ 1),\ Q(\minus{}1,\ 1,\ 0).$ Let $ S$ be the surface generated by rotation the line segment $ PQ$ about $ x$ axis. Answer the following questions. (1) Find the volume of the solid bounded by the surface $ S$ and two planes $ x\equal{}1$ and $ x\equal{}\minus{}1$. (2) Find the cross-section of the solid in (1) by the plane $ y\equal{}0$ to sketch the figure on the palne $ y\equal{}0$. (3) Evaluate the definite integral $ \int_0^1 \sqrt{t^2\plus{}1}\ dt$ by substitution $ t\equal{}\frac{e^s\minus{}e^{\minus{}s}}{2}$. Then use this to find the area of (2).

Suppose $\alpha_i > 0, \beta_i > 0$ for $1 \leq i \leq n, n > 1$ and that \[ \sum^n_{i=1} \alpha_i = \sum^n_{i=1} \beta_i = \pi. \] Prove that \[ \sum^n_{i=1} \frac{\cos(\beta_i)}{\sin(\alpha_i)} \leq \sum^n_{i=1} \cot(\alpha_i). \]

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

Let $ ABC $ be an obtuse triangle with $ \angle A > 90^{\circ} $. Let circle $ O $ be the circumcircle of $ ABC $. $ D $ is a point lying on segment $ AB $ such that $ AD = AC $. Let $ AK $ be the diameter of circle $ O $. Two lines $ AK $ and $ CD $ meet at $ L $. A circle passing through $ D, K, L $ meets with circle $ O $ at $ P ( \ne K ) $ . Given that $ AK = 2, \angle BCD = \angle BAP = 10^{\circ} $, prove that $ DP = \sin ( \frac{ \angle A}{2} )$.

Find the number of all integer-sided [i]isosceles obtuse-angled[/i] triangles with perimeter $ 2008$. [16 points out of 100 for the 6 problems]

Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]

Let $ABCD$ be a square. Consider a variable point $P$ inside the square for which $\angle BAP \ge 60^\circ.$ Let $Q$ be the intersection of the line $AD$ and the perpendicular to $BP$ in $P$. Let $R$ be the intersection of the line $BQ$ and the perpendicular to $BP$ from $C$. [list] [*] [b](a)[/b] Prove that $|BP|\ge |BR|$ [*] [b](b)[/b] For which point(s) $P$ does the inequality in [b](a)[/b] become an equality?[/list]

Let $ABCD$ be an isosceles trapezoid with bases $AB=5$ and $CD=7$ and legs $BC=AD=2 \sqrt{10}.$ A circle $\omega$ with center $O$ passes through $A,B,C,$ and $D.$ Let $M$ be the midpoint of segment $CD,$ and ray $AM$ meet $\omega$ again at $E.$ Let $N$ be the midpoint of $BE$ and $P$ be the intersection of $BE$ with $CD.$ Let $Q$ be the intersection of ray $ON$ with ray $DC.$ There is a point $R$ on the circumcircle of $PNQ$ such that $\angle PRC = 45^\circ.$ The length of $DR$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? [i]Author: Ray Li[/i]

For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?