Let $ABC$ be a scalene triangle. Let $\ell$ be a line passing through point $B$ that lies outside of the triangle $ABC$ and creates different angles with sides $AB$ and $BC$ . The point $M$ is the midpoint of side $AC$ and the ponts $H_a$ and $H_c$ are the bases of the perpendicular lines on the line $\ell$ drawn from points $A$ and $C$ respectively. The circle circumscribing triangle $MBH_a$ intersects AB at the point $A_1$ and the circumscribed circle of triangle $MBH_c$ intersects $BC$ at point $C_1$. The point $A_2$ is symmetric to the point $A$ relative to the point $A_1$ and the point $C_2$ is symmetric to the point $C_1$ relative to the point $C_1$. Prove that the lines $\ell$, $AC_2$ and $CA_2$ intersect at one point. (Yana Kolodach)

In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$

Several figures can be made by attaching two equilateral triangles to the regular pentagon $ ABCDE$ in two of the five positions shown. How many non-congruent figures can be constructed in this way? [asy]unitsize(2cm); pair A=dir(306); pair B=dir(234); pair C=dir(162); pair D=dir(90); pair E=dir(18); draw(A--B--C--D--E--cycle,linewidth(.8pt)); draw(E--rotate(60,D)*E--D--rotate(60,C)*D--C--rotate(60,B)*C--B--rotate(60,A)*B--A--rotate(60,E)*A--cycle,linetype("4 4")); label("$A$",A,SE); label("$B$",B,SW); label("$C$",C,WNW); label("$D$",D,N); label("$E$",E,ENE);[/asy]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

A strip of width $w$ is the set of all points which lie on, or between, two parallel lines distance $w$ apart. Let $S$ be a set of $n$ ($n \ge 3$) points on the plane such that any three different points of $S$ can be covered by a strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.

Isosceles triangles $A_3A_1O_2$ and $A_1A_2O_3$ are constructed on the sides of a triangle $A_1A_2A_3$ as the bases, outside the triangle. Let $O_1$ be a point outside $\Delta A_1A_2A_3$ such that $\angle O_1A_3A_2 =\frac 12\angle A_1O_3A_2$ and $\angle O_1A_2A_3 =\frac 12\angle A_1O_2A_3$. Prove that $A_1O_1\perp O_2O_3$, and if $T$ is the projection of $O_1$ onto $A_2A_3$, then $\frac{A_1O_1}{O_2O_3} = 2\frac{O_1T}{A_2A_3}$.

Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called [i]good [/i] if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ [i]good[/i] triangles. [i]Author: Vyacheslav Yasinskiy, Ukraine[/i]

Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an [i]intersecting set.[/i] Find the maximum possible number of elements of union of two [i]intersecting sets.[/i]

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial? $\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$

Let $ABC$ be a non-degenerate triangle in the euclidean plane. Define a sequence $(C_n)_{n=0}^\infty$ of points as follows: $C_0:=C$, and $C_{n+1}$ is the incenter of the triangle $ABC_n$. Find $\lim_{n\to\infty}C_n$.

Let $ABC$ be a triangle with area $1$ (cm$^2$). Points $D,E$ and $F$ lie on the sides $AB, BC$ and CA, respectively. Prove that $min\{$area of $\vartriangle ADF,$ area of $\vartriangle BED,$ area of $\vartriangle CEF\} \le \frac14$ (cm$^2$).

Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is: $\textbf{(A)}\ xy \qquad\textbf{(B)}\ \frac{y}{x+y} \qquad\textbf{(C)}\ \frac{xy}{x-1} \qquad\textbf{(D)}\ \frac{x+y}{x+1} \qquad\textbf{(E)}\ \frac{x+y}{x-1}$

In the diagram below, points $D, E$, and $F$ are on the inside of equilateral $\vartriangle ABC$ such that $D$ is on $\overline{AE}, E$ is on $\overline{CF}, F$ is on $\overline{BD}$, and the triangles $\vartriangle AEC, \vartriangle BDA$, and $\vartriangle CFB$ are congruent. Given that $AB = 10$ and $DE = 6$, the perimeter of $\vartriangle BDA$ is $\frac{a+b\sqrt{c}}{d}$, where $a, b, c$, and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a + b + c + d$. [img]https://cdn.artofproblemsolving.com/attachments/8/6/98da82fc1c26fa13883a47ba6d45a015622b20.png[/img]

Real numbers $a,b,c$ satisfy $a+b \ne 0$, $b+c \ne 0$ and $c+a \ne 0$. Show that \[\left(\frac{a^2c}{a+b}+\frac{b^2a}{b+c}+\frac{c^2b}{c+a}\right) \cdot \left(\frac{b^2c}{a+b}+\frac{c^2a}{b+c}+\frac{a^2b}{c+a}\right) \ge 0.\]

Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$. Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$. Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=277[/img]

Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$?