Let $x \neq y$ be positive reals satisfying $x^3+2013y=y^3+2013x$, and let $M = \left( \sqrt{3}+1 \right)x + 2y$. Determine the maximum possible value of $M^2$. [i]Proposed by Varun Mohan[/i]

Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.

Let $ABCDEF$ be a convex hexagon, and let $P= AB \cap CD$, $Q = CD \cap EF$, $R = EF \cap AB$, $S = BC \cap DE$, $T = DE \cap FA$, $U = FA \cap BC$. Prove that $\frac{PQ}{CD} = \frac{QR}{EF} = \frac{RP}{AB}$ if and only if $\frac{ST}{DE} = \frac{TU}{FA} = \frac{US}{BC}$

(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB \equal{} a$, $ AD \equal{} b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

In triangle $\vartriangle ABC, D$ and $E$ are midpoints of the sides $BC$ and $AC$, respectively. Lines $AD$ and $BE$ are drawn intersecting at $P$. It turns out that $\angle CAD = 15^o$ and $\angle APB = 60^o$. What is the value of $AB/BC$ ?

Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$. Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. [i]Proposed by Evan Chen[/i]

Let $M$ be the midpoint of a given segment $BC$. Point $A$ is chosen to maximize $\angle ABC$ while subject to the condition that $\angle MAC = 20^o$ . What is the ratio $BC/BA$ ?

$ABCD$ is a trapezoid with $AB || CD$. There are two circles $\omega_1$ and $\omega_2$ is the trapezoid such that $\omega_1$ is tangent to $DA$, $AB$, $BC$ and $\omega_2$ is tangent to $BC$, $CD$, $DA$. Let $l_1$ be a line passing through $A$ and tangent to $\omega_2$(other than $AD$), Let $l_2$ be a line passing through $C$ and tangent to $\omega_1$ (other than $CB$). Prove that $l_1 || l_2$.

If the degree measures of the angles of a triangle are in the ratio $3:3:4$, what is the degree measure of the largest angle of the triangle? $\textbf{(A) }18\qquad\textbf{(B) }36\qquad\textbf{(C) }60\qquad\textbf{(D) }72\qquad\textbf{(E) }90$

Three circles, with radii $p$, $q$ and $r$ and centres $A$, $B$ and $C$ respectively, touch one another externally at points $D$, $E$ and $F$. Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals \[\frac{2pqr}{(p+q)(q+r)(r+p)}.\]

A fenced, rectangular field measures 24 meters by 52 meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the squares must be parallel to the edges of the field. What is the largest number of square test plots into which the field can be partitioned using all or some of the 1994 meters of fence?

A square sheet of paper $ABCD$ is so folded that $B$ falls on the mid point of $M$ of $CD$. Prove that the crease will divide $BC$ in the ration $5 : 3$.

Let $n$ be a five digit number (whose first digit is non-zero) and let $m$ be the four digit number formed from $n$ by removing its middle digit. Determine all $n$ such that $n/m$ is an integer.

The area of $\triangle EBD$ is one third of the area of $3-4-5$ $ \triangle ABC$. Segment $DE$ is perpendicular to segment $AB$. What is $BD$? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,0), B=(5,0), C=(1.8,2.4), D=(5-4sqrt(3)/3,0), E=(5-4sqrt(3)/3,sqrt(3)); pair[] ps={A,B,C,D,E}; draw(A--B--C--cycle); draw(E--D); draw(rightanglemark(E,D,B)); dot(ps); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); label("$3$",midpoint(A--C),NW); label("$4$",midpoint(C--B),NE); label("$5$",midpoint(A--B),SW);[/asy] $ \textbf{(A)}\ \frac{4}{3} \qquad \textbf{(B)}\ \sqrt{5} \qquad \textbf{(C)}\ \frac{9}{4} \qquad \textbf{(D)}\ \frac{4\sqrt{3}}{3} \qquad \textbf{(E)}\ \frac{5}{2} $

Point $P$ is inside $\triangle ABC$. Line segments $APD$, $BPE$, and $CPF$ are drawn with $D$ on $BC$, $E$ on $AC$, and $F$ on $AB$ (see the figure at right). Given that $AP=6$, $BP=9$, $PD=6$, $PE=3$, and $CF=20$, find the area of $\triangle ABC$. [asy] size(200); pair A=origin, B=(7,0), C=(3.2,15), D=midpoint(B--C), F=(3,0), P=intersectionpoint(C--F, A--D), ex=B+40*dir(B--P), E=intersectionpoint(B--ex, A--C); draw(A--B--C--A--D^^C--F^^B--E); pair point=P; 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("$P$", P, dir(0));[/asy]

If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is: $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{2}{3} \qquad\textbf{(C)}\ \frac{3}{4} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{5}$

Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

The base of a triangle is $ 15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. The length of the parallel closer to the base is: $ \textbf{(A)}\ 5\sqrt{6}\text{ inches} \qquad\textbf{(B)}\ 10\text{ inches} \qquad\textbf{(C)}\ 4\sqrt{3}\text{ inches} \qquad\textbf{(D)}\ 7.5\text{ inches}\\ \textbf{(E)}\ \text{none of these}$

The ratio of the measures of two acute angles is $5:4$, and the complement of one of these two angles is twice as large as the complement of the other. What is the sum of the degree measures of the two angles? $\textbf{(A)}\ 75\qquad\textbf{(B)}\ 90\qquad\textbf{(C)}\ 135\qquad\textbf{(D)}\ 150\qquad\textbf{(E)}\ 270$

Points $ B$ and $ C$ lie on $ \overline{AD}$. The length of $ \overline{AB}$ is $ 4$ times the length of $ \overline{BD}$, and the length of $ \overline{AC}$ is $ 9$ times the length of $ \overline{CD}$. The length of $ \overline{BC}$ is what fraction of the length of $ \overline{AD}$? $ \textbf{(A)}\ \frac{1}{36} \qquad \textbf{(B)}\ \frac{1}{13} \qquad \textbf{(C)}\ \frac{1}{10} \qquad \textbf{(D)}\ \frac{5}{36} \qquad \textbf{(E)}\ \frac{1}{5}$

Two circles $ C_1,C_2$ with different radii are given in the plane, they touch each other externally at $ T$. Consider any points $ A\in C_1$ and $ B\in C_2$, both different from $ T$, such that $ \angle ATB \equal{} 90^{\circ}$. (a) Show that all such lines $ AB$ are concurrent. (b) Find the locus of midpoints of all such segments $ AB$.

Let $D$ be a point inside an acute triangle $ABC$ such that $\angle ADC = \angle A +\angle B$, $\angle BDA = \angle B + \angle C$ and $\angle CDB = \angle C + \angle A$. Prove that $\frac{AB \cdot CD}{AD} = \frac{AC \cdot CB} {AB}$.

Let $ABC$ be a triangle with $AB \ne AC$. Let the angle bisector of $\angle BAC$ intersects $BC$ at $P$ and intersects the perpendicular bisector of segment $BC$ at $Q$. Prove that $\frac{PQ}{AQ} =\left( \frac{BC}{AB + AC}\right)^2$

Let $\triangle ABC$ be an equilateral triangle. If $0 < r < 1$, let $D_r$ be the point on $\overline{AB}$ such that $AD_r = r \cdot AB$, let $E_r$ be the point on $\overline{BC}$ such that $BE_r = r \cdot BC$, and let $P_r$ be the point where $\overline{AE_r}$ and $\overline{CD_r}$ intersect. Prove that the set of points $P_r$ (over all $0 < r < 1$) lie on a circle.