In triangle $ ABC$, angle $ C$ is a right angle and $ CB > CA$. Point $ D$ is located on $ \overline{BC}$ so that angle $ CAD$ is twice angle $ DAB$. If $ AC/AD \equal{} 2/3$, then $ CD/BD \equal{} m/n$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$. $ \textbf{(A)}\ 10\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 18\qquad \textbf{(D)}\ 22\qquad \textbf{(E)}\ 26$

The diagonals of trapezoid $ABCD$ with bases $AB$ and $CD$ meet at $P$. Prove the inequality $S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA}$, where $S_{XYZ}$ denotes the area of triangle $XYZ$.

The circles $k_1, k_2, k_3$ with radii ($a>c>b$) $a,b,c$ are tangent to line $d$ at $A,B,C$, respectively. $k_1$ is tangent to $k_2$, and $k_2$ is tangent to $k_3$. The tangent line to $k_3$ at $E$ is parallel to $d$, and it meets $k_1$ at $D$. The line perpendicular to $d$ at $A$ meets line $EB$ at $F$. Prove that $AD=AF$.

Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

A tetrahedron $OA_1B_1C_1$ is given. Let $A_2,A_3 \in OA_1, A_2,A_3 \in OA_1, A_2,A_3 \in OA_1$ be points such that the planes $A_1B_1C_1,A_2B_2C_2$ and $A_3B_3C_3$ are parallel and $OA_1 > OA_2 > OA_3 > 0$. Let $V_i$ be the volume of the tetrahedron $OA_iB_iC_i$ ($i = 1,2,3$) and $V$ be the volume of $OA_1B_2C_3$. Prove that $V_1 +V_2 +V_3 \ge 3V$.

Given 5 points on a plane. Let $\lambda$ be the ratio of maximum value between the points to minimum value between the points. Prove that $\lambda\geq2\sin\frac{3}{10}\pi$.

In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ [b]The segments[/b] $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.

Triangle $ ABC$ has incentre $ I$ and the incircle touches $ BC, CA$ at $ D, E$ respectively. Let $ BI$ meet $ DE$ at $ G$. Show that $ AG$ is perpendicular to $ BG$.

Given an acute-angled triangle $ABC$ ($AB \neq AC$) with orthocenter $H$, circumcenter $O$, and midpoint $M$ of side $BC$. The line $AM$ intersects the circumcircle of triangle $BHC$ at point $K$, with $M$ between $A$ and $K$. The segments $HK$ and $BC$ intersect at point $N$. If $\angle BAM = \angle CAN$, prove that the lines $AN$ and $OH$ are perpendicular.

Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.

In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be $\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$

Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.

Given a straight line $\ell$ and the point $O$ out of the line. Prove that it is possible to move an arbitrary point $A$ in the same plane to the $O$ point, using only rotations around $O$ and symmetry with respect to the $\ell$.

Let $ABC$ be a triangle with $\angle A = 60^\circ$. The point $T$ lies inside the triangle in such a way that $\angle ATB = \angle BTC = \angle CTA = 120^\circ$. Let $M$ be the midpoint of $BC$. Prove that $TA + TB + TC = 2AM$.

Design a planar finite non-empty set $S$ satisfying the following two conditions: (a) every line meets $S$ in at most four points; and (b) every $2$-colouring of $S$ - that is, each point of $S$ is coloured one of two colours - yields (at least) three monochromatic collinear points.

Let $\triangle{ABC}$ be an acute angled triangle. The circle with diameter AB intersects the sides AC and BC at points E and F respectively. The tangents drawn to the circle through E and F intersect at P. Show that P lies on the altitude through the vertex C.

Let $C$ be the part of the graph $y=\frac{1}{x}\ (x>0)$. Take a point $P\left(t,\ \frac{1}{t}\right)\ (t>0)$ on $C$. (i) Find the equation of the tangent $l$ at the point $A(1,\ 1)$ on the curve $C$. (ii) Let $m$ be the line passing through the point $P$ and parallel to $l$. Denote $Q$ be the intersection point of the line $m$ and the curve $C$ other than $P$. Find the coordinate of $Q$. (iii) Express the area $S$ of the part bounded by two line segments $OP,\ OQ$ and the curve $C$ for the origin $O$ in terms of $t$. (iv) Express the volume $V$ of the solid generated by a rotation of the part enclosed by two lines passing through the point $P$ and pararell to the $y$-axis and passing through the point $Q$ and pararell to $y$-axis, the curve $C$ and the $x$-axis in terms of $t$. (v) $\lim_{t\rightarrow 1-0} \frac{S}{V}.$

Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$. Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.

Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$. [img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]

Let $\Delta ABC$ be a scalene triangle. Points $D,E$ lie on side $\overline{AC}$ in the order, $A,E,D,C$. Let the parallel through $E$ to $BC$ intersect $\odot (ABD)$ at $F$, such that, $E$ and $F$ lie on the same side of $AB$. Let the parallel through $E$ to $AB$ intersect $\odot (BDC)$ at $G$, such that, $E$ and $G$ lie on the same side of $BC$. Prove, Points $D,F,E,G$ are concyclic

Let $ ABC $ be a right triangle in $ A, $ and let be a point $ D $ on $ BC. $ The bisectors of $ \angle ADB $ and $ \angle ADC $ intersect $ AB $ and $ AC $ (respectively) in $ M $ and $ N $ (respectively). Show that the small angle between $ BC $ and $ MN $ is equal to $ \frac{1}{2}\cdot\left| \angle ABC -\angle BCA \right| $ if and only if $ D $ is the feet of the perpendicular from $ A. $ [i]Bogdan Enescu[/i]

In an acute triangle $\triangle ABC$, let $A',B',C'$ be the reflection of $A,B,C$ with respect to $BC,CA,AB$. Let $D = B'C \cap BC'$, $E = CA' \cap C'A$, $F = A'B \cap AB'$. Prove that $AD,BE,CF$ are concurrent

In an acute-angled triangle $ABC$ let $BL$ be the bisector, and let $BK$ be the altitude. Let the lines $BL$ and $BK$ meet the circumcircle of $ABC$ again at $W$ and $T$, respectively. Given that $BC = BW$, prove that $TL \perp BC$.

Let $ABC$ be a triangle, let $M$ be the midpoint of $AB$, and let $N$ be the midpoint of $CM$. Let $X$ be a point satisfying both $\angle XMC = \angle MBC$ and $\angle XCM = \angle MCB$ such that $X$ and $B$ lie on opposite sides of $CM$. Let $\omega$ be the circumcircle of triangle $AMX$. $(a)$ Show that $CM$ is tangent to $\omega$. $(b)$ Show that the lines $NX$ and $AC$ intersect on $\omega$