In $\triangle ABC$, three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? [asy] defaultpen(linewidth(0.7)); size(120); pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC; draw(A--B--C--cycle); for(int i = 1; i < 4; ++i) { AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4); draw(AB[i-1] -- AC[i-1]); } filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S);[/asy]

Let $ \omega $ be the incircle of the triangle $ABC$ and with centre $I$. Let $\Gamma $ be the circumcircle of the triangle $AIB$. Circles $ \omega $ and $ \Gamma $ intersect at the point $X$ and $Y$. Let $Z$ be the intersection of the common tangents of the circles $\omega$ and $\Gamma$. Show that the circumcircle of the triangle $XYZ$ is tangent to the circumcircle of the triangle $ABC$.

A rectangle inscribed in a triangle has its base coinciding with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, then: $\textbf{(A)}\ x=\dfrac{1}{2}h \qquad \textbf{(B)}\ x=\dfrac{bh}{b+h} \qquad \textbf{(C)}\ x=\dfrac{bh}{2h+b} \qquad \textbf{(D)}\ x=\sqrt{\dfrac{hb}{2}} \qquad \textbf{(E)}\ x=\dfrac{1}{2}b$

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?

A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.

In triangle $ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is: ${{ \textbf{(A)}\ 1:3 \qquad\textbf{(B)}\ 3:4 \qquad\textbf{(C)}\ 4:3 \qquad\textbf{(D)}\ 3:1 }\qquad\textbf{(E)}\ 7:1 } $

In a convex pentagon every diagonal is parallel to one side. Show that the ratios between the lengths of diagonals and the sides parallel to them are equal and find their value.

Let $ABC$ be a right angled triangle with hypotenuse $AB$ and $P$ be a point on the shorter arc $AC$ of the circumcircle of triangle $ABC$. The line, perpendicuar to $CP$ and passing through $C$, intersects $AP$, $BP$ at points $K$ and $L$ respectively. Prove that the ratio of area of triangles $BKL$ and $ACP$ is independent of the position of point $P$.

The figure shown is the union of a circle and two semicircles of diameters of $ a$ and $ b$, all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is $ \displaystyle \textbf{(A)}\ \sqrt {\frac {a}{b}} \qquad \textbf{(B)}\ \ \frac {a}{b} \qquad \textbf{(C)}\ \ \frac {a^2}{b^2} \qquad \textbf{(D)}\ \ \frac {a \plus{} b}{2b} \qquad \textbf{(E)}\ \ \frac {a^2 \plus{} 2ab}{b^2 \plus{} 2ab}$ [asy]unitsize(2cm); defaultpen(fontsize(10pt)+linewidth(.8pt)); fill(Arc((1/3,0),2/3,0,180)--reverse(Arc((-2/3,0),1/3,180,360))--reverse(Arc((0,0),1,0,180))--cycle,mediumgray); draw(unitcircle); draw(Arc((-2/3,0),1/3,360,180)); draw(Arc((1/3,0),2/3,0,180)); label("$a$",(-2/3,0)); label("$b$",(1/3,0)); draw((-2/3+1/15,0)--(-1/3,0),EndArrow(4)); draw((-2/3-1/15,0)--(-1,0),EndArrow(4)); draw((1/3+1/15,0)--(1,0),EndArrow(4)); draw((1/3-1/15,0)--(-1/3,0),EndArrow(4));[/asy]

Let $z_0+z_1+z_2+\cdots$ be an infinite complex geometric series such that $z_0=1$ and $z_{2013}=\dfrac 1{2013^{2013}}$. Find the sum of all possible sums of this series.

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

The medians of a right triangle $ABC$ ($\angle C = 90^{\circ}$) intersect at $M$. Point $L$ lies on the $AC$ such that $\angle ABL=\angle CBL$. It turned out that $\angle BML = 90^{\circ}$. Find the ration $AB : BC$.

A square is dissected into rectangles with sides parallel to the sides of the square. For each of these rectangles, the ratio of its shorter side to its longer side is considered. Show that the sum of all these ratios is at least $1$.

In the triangle $ABC$, let $D \in (BC)$, $E \in (AB)$, $EF \parallel BC$, $F \in (AC)$, $EG\parallel AD$, $G\in (BC)$ and $M,N$ be the midpoints of $(AD)$ and $(BC)$, respectively. Prove that: a) $\frac{EF}{BC}+\frac{EG}{AD}=1$ b) the midpoint of $[FG]$ lies on the line $ MN$.

Let $ D $ be a point on the side $ BC $ (excluding its endpoints) of a triangle $ ABC $ with $ AB>AC, $ such that $ \frac{\angle BAD}{\angle BAC} $ is a rational number. Prove the following: $$ \frac{\angle BAD}{\angle BAC} < \frac{AB\cdot AC - AC\cdot AD}{AB\cdot AD - AC\cdot AD} $$

A square board is divided by lines parallel to the board sides ($7$ lines in each direction, not necessarily equidistant ) into $64$ rectangles. Rectangles are colored into white and black in alternating order. Assume that for any pair of white and black rectangles the ratio between area of white rectangle and area of black rectangle does not exceed $2.$ Determine the maximal ratio between area of white and black part of the board. White (black) part of the board is the total sum of area of all white (black) rectangles.

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. Find the ratio $L/d$. $ \textbf{(A)}\ 3\sqrt{2}\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 2\sqrt{3}\qquad\textbf{(E)}\ \text{none of the above} $

Let $ 2n$ distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these $ n$ chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)

At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? $\textbf{(A) } 10\%\qquad \textbf{(B) } 12\%\qquad \textbf{(C) } 20\%\qquad \textbf{(D) } 25\%\qquad \textbf{(E) } 33\frac{1}{3}\%$

The fifth and eighth terms of a geometric sequence of real numbers are $ 7!$ and $ 8!$ respectively. What is the first term? $ \textbf{(A)}\ 60\qquad \textbf{(B)}\ 75\qquad \textbf{(C)}\ 120\qquad \textbf{(D)}\ 225\qquad \textbf{(E)}\ 315$

In acute triangle $ ABC,$ points $ P,Q$ lie on its sidelines $ AB,AC,$ respectively. The circumcircle of triangle $ ABC$ intersects of triangle $ APQ$ at $ X$ (different from $ A$). Let $ Y$ be the reflection of $ X$ in line $ PQ.$ Given $ PX>PB.$ Prove that $ S_{\bigtriangleup XPQ}>S_{\bigtriangleup YBC}.$ Where $ S_{\bigtriangleup XYZ}$ denotes the area of triangle $ XYZ.$

Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$. For $n \ge 1$, if $T_n=\triangle ABC$ and $D,E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB,BC$ and $AC$, respectively, then $T_{n+1}$ is a triangle with side lengths $AD,BE,$ and $CF$, if it exists. What is the perimeter of the last triangle in the sequence $(T_n)$? $ \textbf{(A)}\ \frac{1509}{8} \qquad \textbf{(B)}\ \frac{1509}{32} \qquad \textbf{(C)}\ \frac{1509}{64} \qquad \textbf{(D)}\ \frac{1509}{128} \qquad \textbf{(E)}\ \frac{1509}{256} $

Let $\omega$ be the circumcircle of $\triangle{ABC}$. $P$ is an interior point of $\triangle{ABC}$. $A_{1}, B_{1}, C_{1}$ are the intersections of $AP, BP, CP$ respectively and $A_{2}, B_{2}, C_{2}$ are the symmetrical points of $A_{1}, B_{1}, C_{1}$ with respect to the midpoints of side $BC, CA, AB$. Show that the circumcircle of $\triangle{A_{2}B_{2}C_{2}}$ passes through the orthocentre of $\triangle{ABC}$.

Given a parallelogram $ABCD$ with angle $A$ equal to $60^o$. Point $O$ is the the center of a circle circumscribed around triangle $ABD$. Line $AO$ intersects the bisector of the exterior angle $C$ at point $K$. Find the ratio $AO/OK$.