In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$. [i]Proposed by Evan Chen[/i]

A triangle has side lengths 10, 10, and 12. A rectangle has width 4 and area equal to the area of the triangle. What is the perimeter of this rectangle? $ \textbf{(A)}\ 16\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 28\qquad\textbf{(D)}\ 32\qquad\textbf{(E)}\ 36$

An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus? $ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$ [asy]unitsize(1.4cm); defaultpen(linewidth(.8pt)); dotfactor=3; real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points={X,O,Y,Z}; filldraw(Circle(O,r2),mediumgray,black); filldraw(Circle(O,r1),white,black); dot(points); draw(X--Y--O--cycle--Z); label("$O$",O,SSW,fontsize(10pt)); label("$Z$",Z,SW,fontsize(10pt)); label("$Y$",Y,N,fontsize(10pt)); label("$X$",X,NE,fontsize(10pt)); defaultpen(fontsize(8pt)); label("$c$",midpoint(O--Z),W); label("$d$",midpoint(Z--Y),W); label("$e$",midpoint(X--Y),NE); label("$a$",midpoint(X--Z),N); label("$b$",midpoint(O--X),SE);[/asy]

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere? $ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$

In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 12.25 \qquad \textbf{(C)}\ 12.5 \qquad \textbf{(D)}\ 12.75 \qquad \textbf{(E)}\ 13$

Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$, $ AB\equal{}11$, $ BC\equal{}5$, $ CD\equal{}19$, and $ DA\equal{}7$. Bisectors of $ \angle A$ and $ \angle D$ meet at $ P$, and bisectors of $ \angle B$ and $ \angle C$ meet at $ Q$. What is the area of hexagon $ ABQCDP$? $ \textbf{(A)}\ 28\sqrt{3}\qquad \textbf{(B)}\ 30\sqrt{3}\qquad \textbf{(C)}\ 32\sqrt{3}\qquad \textbf{(D)}\ 35\sqrt{3}\qquad \textbf{(E)}\ 36\sqrt{3}$

Isosceles triangle $ABC$ has $AB = AC = 3\sqrt6$, and a circle with radius $5\sqrt2$ is tangent to line $AB$ at $B$ and to line $AC$ at $C$. What is the area of the circle that passes through vertices $A$, $B$, and $C?$ $\textbf{(A) }24\pi\qquad\textbf{(B) }25\pi\qquad\textbf{(C) }26\pi\qquad\textbf{(D) }27\pi\qquad\textbf{(E) }28\pi$

Let $D$ be a point inside the equilateral triangle $\triangle ABC$ such that $|AD|=\sqrt{2}, |BD|=3, |CD|=\sqrt{5}$. $m(\widehat{ADB}) = ?$ $\textbf{(A)}\ 120^{\circ} \qquad\textbf{(B)}\ 105^{\circ} \qquad\textbf{(C)}\ 100^{\circ} \qquad\textbf{(D)}\ 95^{\circ} \qquad\textbf{(E)}\ 90^{\circ}$

Circles of radii 5, 5, 8, and $m/n$ are mutually externally tangent, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

In obtuse triangle $ABC$ with $AB = 7$, $BC = 20$, and $C A = 15$, let point $D$ be the foot of the altitude from $C$ to line $AB$. Evaluate $[ACD]+[BCD]$. (Note that $[XY Z]$ means the area of triangle $XY Z$.) [i]Proposed by Jonathan Liu[/i]

Let $ABC$ be a triangle with incenter $I$. Suppose the reflection of $AB$ across $CI$ and the reflection of $AC$ across $BI$ intersect at a point $X$. Prove that $XI$ is perpendicular to $BC$.

Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$.

Let $ABC$ be a triangle, satisfying $2AC=AB+BC$. If $O$ and $I$ are its circumcenter and incenter, show that $\angle OIB=90^{\circ}$.

In the figure, point $P, Q,R,S$ lies on the side of the rectangle $ABCD$. [img]https://1.bp.blogspot.com/-Ff9rMibTuHA/X9PRPbGVy-I/AAAAAAAAMzA/2ytG0aqe-k0fPL3hbSp_zHrMYAfU-1Y_ACLcBGAsYHQ/s426/2020%2BIndonedia%2BMO%2BProvince%2BP2%2Bq1.png[/img] If it is known that the area of the small square is $1$ unit, determine the area of the rectangle $ABCD$.