What is the total surface area of an ice cream cone, radius $R$, height $H$, with a spherical scoop of ice cream of radius $r$ on top? (Given $R<r$)

Let $\Gamma_1$ and $\Gamma_2$ be two circles with radii $r_1$ and $r_2,$ respectively, where $r_1>r_2.$ Suppose $\Gamma_1$ and $\Gamma_2$ intersect at two distinct points $A$ and $B.$ A point $C$ is selected on ray $\overrightarrow{AB},$ past $B,$ and the tangents to $\Gamma_1$ and $\Gamma_2$ from $C$ are marked as points $P$ and $Q,$ respectively. Suppose that $\Gamma_2$ passes through the center of $\Gamma_1$ and that points $P, B, Q$ are collinear in that order, with $PB=3$ and $QB=2.$ What is the length of $AB?$ [i]Proposed by Kyle Lee[/i]

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\neq X$ and $B\neq X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.

The figure shown is called a [i]trefoil[/i] and is constructed by drawing circular sectors about sides of the congruent equilateral triangles. What is the area of a trefoil whose horizontal base has length $ 2$? [asy]unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(12pt)); pair O=(0,0), A=dir(0), B=dir(60), C=dir(120), D=dir(180); pair E=B+C; draw(D--E--B--O--C--B--A,linetype("4 4")); draw(Arc(O,1,0,60),linewidth(1.2pt)); draw(Arc(O,1,120,180),linewidth(1.2pt)); draw(Arc(C,1,0,60),linewidth(1.2pt)); draw(Arc(B,1,120,180),linewidth(1.2pt)); draw(A--D,linewidth(1.2pt)); draw(O--dir(40),EndArrow(HookHead,4)); draw(O--dir(140),EndArrow(HookHead,4)); draw(C--C+dir(40),EndArrow(HookHead,4)); draw(B--B+dir(140),EndArrow(HookHead,4)); label("2",O,S); draw((0.1,-0.12)--(1,-0.12),EndArrow(HookHead,4),EndBar); draw((-0.1,-0.12)--(-1,-0.12),EndArrow(HookHead,4),EndBar);[/asy]$ \textbf{(A)}\ \frac13\pi\plus{}\frac{\sqrt3}{2} \qquad \textbf{(B)}\ \frac23\pi \qquad \textbf{(C)}\ \frac23\pi\plus{}\frac{\sqrt3}{4} \qquad \textbf{(D)}\ \frac23\pi\plus{}\frac{\sqrt3}{3} \qquad \textbf{(E)}\ \frac23\pi\plus{}\frac{\sqrt3}{2}$

Let $O$ be the circumcenter of a triangle $ABC$, and let $l$ be the line going through the midpoint of the side $BC$ and is perpendicular to the bisector of $\angle BAC$. Determine the value of $\angle BAC$ if the line $l$ goes through the midpoint of the line segment $AO$.

In the coordinate plane, denote by $ S(a)$ the area of the region bounded by the line passing through the point $ (1,\ 2)$ with the slope $ a$ and the parabola $ y\equal{}x^2$. When $ a$ varies in the range of $ 0\leq a\leq 6$, find the value of $ a$ such that $ S(a)$ is minimized.

Given a triangle with $a,b,c$ sides and with the area $1$ ($a \ge b \ge c$). Prove that $b^2 \ge 2$.

$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.

Let $CD$ be an altitude of right-angled triangle $ABC$ with $\angle C = 90$. Regular triangles$ AED$ and $CFD$ are such that $E$ lies on the same side from $AB$ as $C$, and $F$ lies on the same side from $CD$ as $B$. The line $EF$ meets $AC$ at $L$. Prove that $FL = CL + LD$

Let $ABC$ be a isosceles triangle, with $AB=AC$. A line $r$ that pass through the incenter $I$ of $ABC$ touches the sides $AB$ and $AC$ at the points $D$ and $E$, respectively. Let $F$ and $G$ be points on $BC$ such that $BF=CE$ and $CG=BD$. Show that the angle $\angle FIG$ is constant when we vary the line $r$.

Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.

Altitudes $AH_A, BH_B, CH_C$ of triangle $ABC$ intersect at $H$, and let $M$ be the midpoint of the side $AC$. The bisector $BL$ of $\triangle ABC$ intersects $H_AH_C$ at point $K$. The line through $L$ parallel to $HM$ intersects $BH_B$ in point $T$. Prove that $TK = TL$. [i]Proposed by Anton Trygub[/i]

Two squares are arranged as shown in the picture. Prove that the areas of shaded quadrilaterals are equal. [img]https://3.bp.blogspot.com/-W50DOuizFvY/XT6wh3-L6sI/AAAAAAAAKaw/pIW2RKmttrwPAbrKK3bpahJz7hfIZwM8QCK4BGAYYCw/s400/Oral%2BSharygin%2B2016%2B10.11%2Bp3.png[/img]

Each side of a triangle $ABC$ has been divided into $n+1$ equal parts. Find the number of triangles with the vertices at the division points having no side parallel to or lying at a side of $\triangle ABC$.

A circle $\omega$ is strictly inside triangle $ABC$. The tangents from $A$ to $\omega$ intersect $BC$ in $A_1,A_2$ define $B_1,B_2,C_1,C_2$ similarly. Prove that if five of six points $A_1,A_2,B_1,B_2,C_1,C_2$ lie on a circle the sixth one lie on the circle too.

The point $ P$ is a fixed point on a circle and $ Q$ is a fixed point on a line. The point $ R$ is a variable point on the circle such that $ P,Q,$ and $ R$ are not collinear. The circle through $ P,Q,$ and $ R$ meets the line again at $ V$. Show that the line $ VR$ passes through a fixed point.

A straight line is drawn across an $8\times 8$ chessboard. It is said to [i]pierce [/i]a square if it passes through an interior point of the square. At most how many of the $64$ squares can this line [i]pierce[/i]?

Point $D$ is on the hypotenuse $BC$ of right-angled triangle $ABC$. The inradii of triangles $ADB$ and $ADC$ are equal. Prove that $S_{ABC} = AD^2$, where $S$ is the area function.

3.Let ABCD be a parallelogram with side AB longer than AD and acute angle $\angle DAB$. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.

Circles $S,S_1,S_2$ are given in a plane. $S_1$ and $S_2$ touch each other externally, and both touch $S$ internally at $A_1$ and $A_2$ respectively. The common internal tangent to $S_1$ and $S_2$ meets $S$ at $P$ and $Q.$ Let $B_1$ and $B_2$ be the intersections of $PA_1$ and $PA_2$ with $S_1$ and $S_2$, respectively. Prove that $B_1B_2$ is a common tangent to $S_1,S_2$

Equilateral triangles $\triangle{ABC}$ and $\triangle{DEF}$ are drawn such that points $B, E, F,$ and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle{ABC}.$ If $BE=14, EF=15,$ and $FC=16,$ compute $AD.$

In a triangle $ABC$, the bisector of the largest angle $\angle A$ meets $BC$ at point $D$. Let $E$ and $F$ be the feet of perpendiculars from $D$ to $AC$ and $AB$, respectively. Let $R$ denote the ratio between the areas of triangles $DEB$ and $DFC$. (a) Prove that, for every real number $r > 0$, one can construct a triangle ABC for which $R$ is equal to $r$. (b) Prove that if $R$ is irrational, then at least one side length of $\vartriangle ABC$ is irrational. (c) Give an example of a triangle $ABC$ with exactly two sides of irrational length, but with rational $R$.

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.