Let $ABC$ be a triangle. Suppose that the perpendicular bisector of $BC$ meets the circle of diameter $AB$ at a point $D$ at the opposite side of $BC$ with respect to $A$, and meets the circle through $A, C, D$ again at $E$. Prove that $\angle ACE=\angle BCD$. [i]Proposed by José Manuel Guerra and Victor Domínguez[/i]

The circle $\Gamma$ through $A$ of triangle $ABC$ meets sides $AB,AC$ at $E$,$F$ respectively, and circumcircle of $ABC$ at $P$. Prove: Reflection of $P$ across $EF$ is on $BC$ if and only if $\Gamma$ passes through $O$ (the circumcentre of $ABC$).

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Let $ABCD$ be a convex quadrileteral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. What is $m(\widehat{CAD})$? $ \textbf{(A)}\ 25^\circ \qquad\textbf{(B)}\ 30^\circ \qquad\textbf{(C)}\ 35^\circ \qquad\textbf{(D)}\ 38^\circ \qquad\textbf{(E)}\ 40^\circ $

In parallellogram $ABCD$, on the arc $BC$ of the circumcircle $(ABC)$, not containing the point $A$, we take a point $P$ and on the $[AC$, we take a point $Q$ such that $\angle PBC= \angle CDQ$. Prove that $(APQ)$ is tangent to $AB$.

Let $DFGE$ be a cyclic quadrilateral. Line $DF$ intersects $EG$ at $C,$ and line $FE$ intersects $DG$ at $H.$ $J$ is the midpoint of $FG.$ The line $\ell$ is the reflection of the line $DE$ in $CH,$ and it intersects line $GF$ at $I.$ Prove that $C,J,H,I$ are concyclic.

$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$. $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$. Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$, and $r$ meets $A_{1}B_{1}$ at $C_{2}$. Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$, and that $X$ lies on the circumcircle of $ABC$.

Triangle $ ABC$ has $ AB \equal{} 2 \cdot AC$. Let $ D$ and $ E$ be on $ \overline{AB}$ and $ \overline{BC}$, respectively, such that $ \angle{BAE} \equal{} \angle{ACD}.$ Let $ F$ be the intersection of segments $ AE$ and $ CD$, and suppose that $ \triangle{CFE}$ is equilateral. What is $ \angle{ACB}$? $ \textbf{(A)}\ 60^{\circ}\qquad \textbf{(B)}\ 75^{\circ}\qquad \textbf{(C)}\ 90^{\circ}\qquad \textbf{(D)}\ 105^{\circ}\qquad \textbf{(E)}\ 120^{\circ}$

Find the locus of points $P$ in an equilateral triangle $ABC$ for which the square root of the distance of $P$ to one of the sides is equal to the sum of the square root of the distance of $P$ to the two other sides.

Let $BC$ and $BD$ be the tangent lines to the circle with diameter $AC$. Let $E$ be the second point of intersection of line $CD$ and the circumscribed circle of triangle $ABC$. Prove that $CD= 2DE$. (Matthew Kurskyi)

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

In acute triangle $ABC$, let $\odot O$ be its circumcircle, $\odot I$ be its incircle. Tangents at $B,C$ to $\odot O$ meet at $L$, $\odot I$ touches $BC$ at $D$. $AY$ is perpendicular to $BC$ at $Y$, $AO$ meets $BC$ at $X$, and $OI$ meets $\odot O$ at $P,Q$. Prove that $P,Q,X,Y$ are concyclic if and only if $A,D,L$ are collinear.

Let $T$ be a regular tetrahedron. Let $t$ be the regular tetrahedron whose vertices are the centers of the faces of $T$. Let $O$ be the circumcenter of either tetrahedron. Given a point $P$ different from $O$, let $m(P)$ be the midpoint of the points of intersection of the ray $\overrightarrow{OP}$ with $t$ and $T$. Let $S$ be the set of eight points $m(P)$ where $P$ is a vertex of either $t$ or $T$. What is the volume of the convex hull of $S$ divided by the volume of $t$?

An ellipse lies in the $xy$-plane and is tangent to both the $x$-axis and $y$-axis. Given that one of the foci is at $(9, 12)$, compute the minimum possible distance between the two foci.

The point $E$ lies on the base $[AD]$ of the trapezoid $ABCD$. The perimeters of the triangles $ABE, BCE$ and $CDE$ are equal. Prove that $|BC| = |AD|/2$

Let $A$, $B$, $C$, $D$ be four points on a circle in that order. Also, $AB=3$, $BC=5$, $CD=6$, and $DA=4$. Let diagonals $AC$ and $BD$ intersect at $P$. Compute $\frac{AP}{CP}$.

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

Let $ ABC$ be a triangle with $ \measuredangle{BAC} < \measuredangle{ACB}$. Let $ D$, $ E$ be points on the sides $ AC$ and $ AB$, such that the angles $ ACB$ and $ BED$ are congruent. If $ F$ lies in the interior of the quadrilateral $ BCDE$ such that the circumcircle of triangle $ BCF$ is tangent to the circumcircle of $ DEF$ and the circumcircle of $ BEF$ is tangent to the circumcircle of $ CDF$, prove that the points $ A$, $ C$, $ E$, $ F$ are concyclic. [i]Author: Cosmin Pohoata[/i]

The line segments $AB$ and $CD$ are perpendicular and intersect at point $X$. Additionally, the following equalities hold: $AC=BD$, $AD=BX$, and $DX=1$. Determine the length of segment $CX$.

Find the side length of the largest square that can be inscribed in the unit cube.

Point $P$ is chosen inside triangle $ABC$ so that $\angle APC+\angle ABC=180^o$ and $BC=AP.$ On the side $AB$, a point $K$ is chosen such that $AK = KB + PC$. Prove that $CK \perp AB$.

Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O$, and that the ratio of the volume of $O$ to that of $C$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m+n$.

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

Daniel finds a rectangular index card and measures its diagonal to be 8 centimeters. Daniel then cuts out equal squares of side 1 cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area of the original index card? [asy] unitsize(0.6 cm); pair A, B, C, D, E, F, G, H; real x, y; x = 9; y = 5; A = (0,y); B = (x - 1,y); C = (x - 1,y - 1); D = (x,y - 1); E = (x,0); F = (1,0); G = (1,1); H = (0,1); draw(A--B--C--D--E--F--G--H--cycle); draw(interp(C,G,0.03)--interp(C,G,0.97), dashed, Arrows(6)); draw(interp(A,E,0.03)--interp(A,E,0.97), dashed, Arrows(6)); label("$1$", (B + C)/2, W); label("$1$", (C + D)/2, S); label("$8$", interp(A,E,0.3), NE); label("$4 \sqrt{2}$", interp(G,C,0.2), SE); [/asy] $\textbf{(A) }14\qquad\textbf{(B) }10\sqrt{2}\qquad\textbf{(C) }16\qquad\textbf{(D) }12\sqrt{2}\qquad\textbf{(E) }18$

The triangle $ABC$ is drawn on the board such that $AB + AC = 2BC$. The bisectors $AL_1, BL_2, CL_3$ were drawn in this triangle, after which everything except the points $L_1, L_2, L_3$ was erased. Use a compass and a ruler to reconstruct triangle $ABC$.