Let $ABC$ be an acute scalene triangle. Let $D, E$ be points on segments $AB, AC$ respectively, such that $BD=CE$. Prove that the nine-point centers of $ADE$, $ACD$, $ABC$, $AEB$ form a rhombus. [i]Proposed by Malay Mahajan and Siddharth Choppara[/i]

Let $P$ be a point outside a circle $\omega$. The tangents from $P$ to $\omega$ are drawn touching $\omega$ at points $A$ and $B$. Let $M$ and $N$ be the midpoints of $AP$ and $AB$, respectively. Line $MN$ is extended to cut $\omega$ at $C$ so that $N$ lies between $M$ and $C$. Line $PC$ intersects $\omega$ again at $D$, and lines $ND$ and $PB$ intersect at $O$. Prove that $MNOP$ is a rhombus.

Say that a convex quadrilateral is [i]tasty[/i] if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.

Let $ABCD$ be a convex quadrilateral. Prove that the incircles of the triangles $ABC$, $BCD$, $CDA$ and $DAB$ have a point in common if, and only if, $ABCD$ is a rhombus.

$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.

The points $A,B,C$ are in this order on line $D$, and $AB = 4BC$. Let $M$ be a variable point on the perpendicular to $D$ through $C$. Let $MT_1$ and $MT_2$ be tangents to the circle with center $A$ and radius $AB$. Determine the locus of the orthocenter of the triangle $MT_1T_2.$

Given a circle $\Gamma$ and the positive numbers $h$ and $m$, construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude $h$ and the sum of its parallel sides is $m$.

Four circles of radius $ 1$ are each tangent to two sides of a square and externally tangent to a circle of radius $ 2$, as shown. What is the area of the square? [asy]unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real h=3*sqrt(2)/2; pair O0=(0,0), O1=(h,h), O2=(-h,h), O3=(-h,-h), O4=(h,-h); pair X=O0+2*dir(30), Y=O2+dir(45); draw((-h-1,-h-1)--(-h-1,h+1)--(h+1,h+1)--(h+1,-h-1)--cycle); draw(Circle(O0,2)); draw(Circle(O1,1)); draw(Circle(O2,1)); draw(Circle(O3,1)); draw(Circle(O4,1)); draw(O0--X); draw(O2--Y); label("$2$",midpoint(O0--X),NW); label("$1$",midpoint(O2--Y),SE);[/asy]$ \textbf{(A)}\ 32 \qquad \textbf{(B)}\ 22 \plus{} 12\sqrt {2}\qquad \textbf{(C)}\ 16 \plus{} 16\sqrt {3}\qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 36 \plus{} 16\sqrt {2}$

Given four points $ A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that \[ A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, \] denote by $ O_i$ the circumcenter of $ \triangle A_j A_k A_l$ with $ \{i,j,k,l\} \equal{} \{1,2,3,4\}.$ Assuming $ \forall i A_i \neq O_i ,$ prove that the four lines $ A_iO_i$ are concurrent or parallel. [i]Nikolai Ivanov Beluhov, Bulgaria[/i]

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy: $$ MA\cdot MC+MB\cdot MD=AB^2 $$ [i]Ovidiu Pop[/i]

Let $ABCD$ be a convex quadrilateral. Suppose that similar isosceles triangles $APB, BQC, CRD, DSA$ with the bases on the sides of $ABCD$ are constructed in the exterior of the quadrilateral such that $PQRS$ is a rectangle but not a square. Show that $ABCD$ is a rhombus.

What is the area enclosed by the graph of $ |3x| \plus{} |4y| \equal{} 12$? $ \textbf{(A)}\ 6\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 24\qquad \textbf{(E)}\ 25$

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

Rhombus $ ABCD$, with a side length $ 6$, is rolled to form a cylinder of volume $ 6$ by taping $ \overline{AB}$ to $ \overline{DC}.$ What is $ \sin(\angle ABC)$? $ \textbf{(A)}\ \frac {\pi}{9} \qquad \textbf{(B)}\ \frac {1}{2} \qquad \textbf{(C)}\ \frac {\pi}{6} \qquad \textbf{(D)}\ \frac {\pi}{4} \qquad \textbf{(E)}\ \frac {\sqrt3}{2}$

On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that: (a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$; (b) $P,A,Q$ are collinear. [i]Virginia Tica, Vasile Tica[/i]