In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

There are 6 distinct values of $x$ strictly between $0$ and $\frac{\pi}{2}$ that satisfy the equation \[ \tan(15 x) = 15 \tan(x) . \] Call these 6 values $r_1$, $r_2$, $r_3$, $r_4$, $r_5$, and $r_6$. What is the value of the sum \[ \frac{1}{\tan^2 r_1} + \frac{1}{\tan^2 r_2} + \frac{1}{\tan^2 r_3} + \frac{1}{\tan^2 r_4} + \frac{1}{\tan^2 r_5} + \frac{1}{\tan^2 r_6} \, ? \]

Given that circle $ \omega$ is tangent internally to circle $ \Gamma$ at $ S.$ $ \omega$ touches the chord $ AB$ of $ \Gamma$ at $ T$. Let $ O$ be the center of $ \omega.$ Point $ P$ lies on the line $ AO.$ Show that $ PB\perp AB$ if and only if $ PS\perp TS.$

A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$. If $ \angle A\equal{}\angle B\equal{}120^{\circ}, \angle D\equal{}90^{\circ}$ and $ BC\equal{}1$, find, with proof, the length of $ AD$.

Let $k$ be the circle and $A$ and $B$ points on circle which are not diametrically opposite. On minor arc $AB$ lies point arbitrary point $C$. Let $D$, $E$ and $F$ be foots of perpendiculars from $C$ on chord $AB$ and tangents of circle $k$ in points $A$ and $B$. Prove that $CD= \sqrt {CE \cdot CF}$

Evaluate $ \int_0^{\frac{\pi}{4}} \frac{1}{\cos \theta}\sqrt{\frac{1\plus{}\sin \theta}{\cos \theta}}\ d\theta$.

Convex quadrilateral $ ABCD$ is inscribed in a circle, $ \angle{A}\equal{}60^o$, $ BC\equal{}CD\equal{}1$, rays $ AB$ and $ DC$ intersect at point $ E$, rays $ BC$ and $ AD$ intersect each other at point $ F$. It is given that the perimeters of triangle $ BCE$ and triangle $ CDF$ are both integers. Find the perimeter of quadrilateral $ ABCD$.

Let a triangle $ABC$ satisfy $AC = BC$; in other words, let $ABC$ be an isosceles triangle with base $AB$. Let $P$ be a point inside the triangle $ABC$ such that $\angle PAB = \angle PBC$. Denote by $M$ the midpoint of the segment $AB$. Show that $\angle APM + \angle BPC = 180^{\circ}$.

Prove that for every real number $x$ and positive integer $n$ $$|\cos x|+|\cos2x|+|\cos2^2x|+\ldots+|\cos2^nx|\ge\frac n{2\sqrt2}.$$

Prove that the function \[ f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}\] (where the positive value of the square root is taken) is monotonically decreasing in the interval $ 0<\nu<1$. [P. Turan]

Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.

What is the value of the sum \[ \sum_z \frac{1}{{\left|1 - z\right|}^2} \, , \] where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^2 A + \sin ^2 B + \sin^2 C = 2 \left( \cos ^2 A + \cos ^2 B + \cos ^2 C \right),$$ prove that the triangle must have a right angle.

On the domain $0\leq \theta \leq 2\pi:$ (a) Prove that $\sin^{2}\theta \cdot \sin 2\theta$ takes its maximum at $\frac{\pi}{3}$ and $\frac{4 \pi}{3}$ (and hence its minimum at $\frac{2 \pi}{3}$ and $\frac{ 5 \pi}{3}$). (b) Show that $$| \sin^{2} \theta \cdot \sin^{3} 2\theta \cdot \sin^{3} 4 \theta \cdots \sin^{3} 2^{n-1} \theta \cdot \sin 2^{n} \theta |$$ takes its maximum at $\frac{4 \pi}{3}$ (the maximum may also be attained at other points). (c) Derive the inequality: $$ \sin^{2} \theta \cdot \sin^{2} 2\theta \cdots \sin^{2} 2^{n} \theta \leq \left( \frac{3}{4} \right)^{n}.$$

Circle $I$ is the incircle of $\triangle ABC$. Circle $I$ is tangent to sides $BC$ and $AC$ at $M,N$ respectively. $E,F$ are midpoints of sides $AB$ and $AC$ respectively. Lines $EF, BI$ intersect at $D$. Show that $M,N,D$ are collinear.

In a triangle $ABC$, let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$. If $IM$ = $IA$, find the least possible measure of $\angle{AIM}$.

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$? $ \textbf{(A)}\ \sqrt{10} \qquad \textbf{(B)}\ \frac{7}{2} \qquad \textbf{(C)}\ \sqrt{14} \qquad \textbf{(D)}\ \sqrt{15} \qquad \textbf{(E)}\ 4$

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

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 \[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]

Prove that if $A,B,$ and $C$ are angles of a triangle measured in radians then $A \cos B +\sin A \cos C >0.$

Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula \[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]

How many values of $x$, $-19<x<98$, satisfy $\cos^2 x + 2\sin^2 x = 1?$