Let $\triangle{ABC}$ be a triangle with circumcircle $\tau$. The tangentlines to $\tau$ through $A$ and $B$ intersect at $T$. The circle through $A$, $B$ and $T$ intersects $BC$ and $AC$ again at $D$ and $E$, respectively; $CT$ and $BE$ intersect at $F$. Suppose $D$ is the midpoint of $BC$. Calculate the ratio $BF:BE$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 5)[/i]

Two geometric sequences $ a_1,a_2,a_3,\ldots$ and $b_1,b_2,b_3\ldots $have the same common ratio, with $a_1=27$,$b_1=99$, and $a_{15}=b_{11}$. Find $a_9.$

Consider a cube $ABCDA'B'C'D$ (with $AB\perp AA'\parallel BB'\parallel CC'\parallel DD$). Let $X$ be an inner point of the segment $AB$ and denote $Y$ the intersection of the edge $AD$ and the plane $B'D'X$. (a) Let $M=B'Y\cap D'X$. Find the locus of all $M$s. (b) Determine whether there is a quadrilateral $B'D'YX$ such that its diagonals divide each other in the ratio 1:2.

What is the area of the region defined by $x^2+3y^2 \le 4$ and $y^2+3x^2 \le 4$?

$ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. The ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$ is [asy] size(170); defaultpen(linewidth(0.7)+fontsize(10)); pair H=origin, B=(1,-(1/sqrt(3))), C=(-1,-(1/sqrt(3))), A=(0,(2/sqrt(3))), E=(2,-(2/sqrt(3))), F=(-2,-(2/sqrt(3))), D=(0,(4/sqrt(3))); draw(A--B--C--A^^D--E--F--D); label("$A'$", A, dir(90)); label("$B'$", B, SE); label("$C'$", C, SW); label("$A$", D, dir(90)); label("$B$", E, dir(0)); label("$C$", F, W); [/asy] $ \textbf{(A)}\ \frac{1}{36}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{\sqrt{3}}{4}\qquad\textbf{(E)}\ \frac{9+8\sqrt{3}}{36} $

in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$. a) in what ratio does $EF$ cut the digonals?(13 points) b) find $\frac{AF}{FD}$.(5 points)

Prove that in all triangles $ABC$ with $\angle A = 2\angle B$ the distance from $C$ to $A$ and to the perpendicular bisector of $AB$ are in the same ratio.

Let $ABC$ be a triangle. Prove that there is a line $\ell$ (in the plane of triangle $ABC$) such that the intersection of the interior of triangle $ABC$ and the interior of its reflection $A'B'C'$ in $\ell$ has area more than $\frac23$ the area of triangle $ABC$.

Given a triangle $ABC$, let $A',B',C'$ be the perpendicular feet dropped from the centroid $G$ of the triangle $ABC$ onto the sides $BC,CA,AB$ respectively. Reflect $A',B',C'$ through $G$ to $A'',B'',C''$ respectively. Prove that the lines $AA'',BB'',CC''$ are concurrent.

The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion $\text{(A)} \ x: y: z ~~\text{(B)} \ z: y: x ~~ \text{(C)} \ y: z: x~~ \text{(D)} \ yz: xz: xy ~~ \text{(E)} \ xz: yx: zy$

Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid. [i]Author: A. Akopyan, A. Myakishev[/i]

Circles of diameter 1 inch and 3 inches have the same center. The smaller circle is painted red, and the portion outside the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? $ \textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9$

Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of the circumcircle at point $K$. Prove that the ratio $AK:BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

In this figure $\angle RFS = \angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: [asy] import olympiad; pair F,R,S,D; F=origin; R=5*dir(aCos(9/16)); S=(7.5,0); D=4*dir(aCos(9/16)+aCos(1/8)); label("$F$",F,SW);label("$R$",R,N); label("$S$",S,SE); label("$D$",D,W); label("$7\frac{1}{2}$",(F+S)/2.5,SE); label("$4$",midpoint(F--D),SW); label("$5$",midpoint(F--R),W); label("$6$",midpoint(D--R),N); draw(F--D--R--F--S--R); markscalefactor=0.1; draw(anglemark(S,F,R)); draw(anglemark(F,D,R)); //Credit to throwaway1489 for the diagram[/asy] $\textbf{(A)}\ \text{undetermined} \qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\dfrac{1}{2} \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 6\dfrac{1}{4}$