Given triangle $ ABC$. Let the tangent lines of the circumcircle of $ AB$ at $ B$ and $ C$ meet at $ A_0$. Define $ B_0$ and $ C_0$ similarly. a) Prove that $ AA_0,BB_0,CC_0$ are concurrent. b) Let $ K$ be the point of concurrency. Prove that $ KG\parallel BC$ if and only if $ 2a^2\equal{}b^2\plus{}c^2$.

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

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

Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter?

Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $ [i]Dorin Andrica[/i]

Let triangle $ABC$ be such that its circumradius is $R = 1.$ Let $r$ be the inradius of $ABC$ and let $p$ be the inradius of the orthic triangle $A'B'C'$ of triangle $ABC.$ Prove that \[ p \leq 1 - \frac{1}{3 \cdot (1+r)^2}. \] [hide="Similar Problem posted by Pascual2005"] Let $ABC$ be a triangle with circumradius $R$ and inradius $r$. If $p$ is the inradius of the orthic triangle of triangle $ABC$, show that $\frac{p}{R} \leq 1 - \frac{\left(1+\frac{r}{R}\right)^2}{3}$. [i]Note.[/i] The orthic triangle of triangle $ABC$ is defined as the triangle whose vertices are the feet of the altitudes of triangle $ABC$. [b]SOLUTION 1 by mecrazywong:[/b] $p=2R\cos A\cos B\cos C,1+\frac{r}{R}=1+4\sin A/2\sin B/2\sin C/2=\cos A+\cos B+\cos C$. Thus, the ineqaulity is equivalent to $6\cos A\cos B\cos C+(\cos A+\cos B+\cos C)^2\le3$. But this is easy since $\cos A+\cos B+\cos C\le3/2,\cos A\cos B\cos C\le1/8$. [b]SOLUTION 2 by Virgil Nicula:[/b] I note the inradius $r'$ of a orthic triangle. Must prove the inequality $\frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ From the wellknown relations $r'=2R\cos A\cos B\cos C$ and $\cos A\cos B\cos C\le \frac 18$ results $\frac{r'}{R}\le \frac 14.$ But $\frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longleftrightarrow \frac 13\left( 1+\frac rR\right)^2\le \frac 34\Longleftrightarrow$ $\left(1+\frac rR\right)^2\le \left(\frac 32\right)^2\Longleftrightarrow 1+\frac rR\le \frac 32\Longleftrightarrow \frac rR\le \frac 12\Longleftrightarrow 2r\le R$ (true). Therefore, $\frac{r'}{R}\le \frac 14\le 1-\frac 13\left( 1+\frac rR\right)^2\Longrightarrow \frac{r'}{R}\le 1-\frac 13\left( 1+\frac rR\right)^2.$ [b]SOLUTION 3 by darij grinberg:[/b] I know this is not quite an ML reference, but the problem was discussed in Hyacinthos messages #6951, #6978, #6981, #6982, #6985, #6986 (particularly the last message). [/hide]

[1] Tokyo Univ. of Science: $\int \frac{\ln x}{(x+1)^2}dx$ [2] Saitama Univ.: $\int \frac{5}{3\sin x+4\cos x}dx$ [3] Yokohama City Univ.: $\int_1^{\sqrt{3}} \frac{1}{\sqrt{x^2+1}}dx$ [4] Daido Institute of Technology: $\int_0^{\frac{\pi}{2}} \frac{\sin ^ 3 x}{\sin x +\cos x}dx$ [5] Gunma Univ.: $\int_0^{\frac{3\pi}{4}} \{(1+x)\sin x+(1-x)\cos x\}dx$

Find $ \lim_{n\to\infty} \int_0^{\frac{\pi}{2}} x|\cos (2n\plus{}1)x|\ dx$.

For each integer $n > 0$ show that there is a polynomial $p(x)$ such that $p(2 cos x) = 2 cos nx$.

Suppose $x,y,z$, and $w$ are positive reals such that \[ x^2 + y^2 - \frac{xy}{2} = w^2 + z^2 + \frac{wz}{2} = 36 \] \[ xz + yw = 30. \] Find the largest possible value of $(xy + wz)^2$. [i]Author: Alex Zhu[/i]

Is there an infinite set of points in the plane such that no three points are collinear, and the distance between any two points is rational?

Let $ABCD$ be a rectangle with area $1$, and let $E$ lie on side $CD$. What is the area of the triangle formed by the centroids of triangles $ABE$, $BCE$, and $ADE$?

A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

What is the value of $ \sum_{n=1}^\infty (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$ for all $a$ and $b$ in the interval $(0, \pi/2)$ .

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

Point $G$ is where the medians of the triangle $ABC$ intersect and point $D$ is the midpoint of side $BC$. The triangle $BDG$ is equilateral with side length 1. Determine the lengths, $AB$, $BC$, and $CA$, of the sides of triangle $ABC$. [asy] size(200); defaultpen(fontsize(10)); real r=100.8933946; pair A=sqrt(7)*dir(r), B=origin, C=(2,0), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C); draw(A--B--C--A--D^^B--E^^C--F); pair point=G; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$G$", G, dir(20)); label("1", B--G, dir(150)); label("1", D--G, dir(30)); label("1", B--D, dir(270));[/asy]

Given triangle $ ABC$ of area 1. Let $ BM$ be the perpendicular from $ B$ to the bisector of angle $ C$. Determine the area of triangle $ AMC$.

In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear. [i]proposed by Amirhossein Zabeti[/i]

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]

If an angle of a triangle remains unchanged but each of its two including sides is doubled, then the area is multiplied by: $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ \text{more than }6$

In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.