Harvard-MIT Mathematics Tournament 2014-20 (HMMT) 70p

geometry problems from Harvard-MIT Mathematics Tournament (HMMT)
with aops links in the names

collected pdfs  

Harvard-MIT Mathematics Tournament (HMMT) 1998 - 2006 with solutions

Harvard-MIT Mathematics Tournament (HMMT) 2007 - 17 with solutions

under construction

Harvard-MIT Mathematics Tournament Invitational Competition (HMIC)

2014 - 2020 geometry rounds

collected inside aops here

2014 HMMT February Geometry 1
Let $O_1$ and $O_2$ be concentric circles with radii 4 and 6, respectively. A chord $AB$ is drawn in $O_1$ with length $2$. Extend $AB$ to intersect $O_2$ in points $C$ and $D$. Find $CD$.

2014 HMMT February Geometry 2
Point $P$ and line $\ell$ are such that the distance from $P$ to $\ell$ is $12$. Given that $T$ is a point on $\ell$ such that $PT = 13$, find the radius of the circle passing through $P$ and tangent to $\ell$ at $T$.

2014 HMMT February Geometry 3
$ABC$ is a triangle such that $BC = 10$, $CA = 12$. Let $M$ be the midpoint of side $AC$. Given that $BM$ is parallel to the external bisector of $\angle A$, find area of triangle $ABC$. (Lines $AB$ and $AC$ form two angles, one of which is $\angle BAC$. The external angle bisector of $\angle A$ is the line that bisects the other angle.

2014 HMMT February Geometry 4
In quadrilateral $ABCD$, $\angle DAC = 98^{\circ}$, $\angle DBC = 82^\circ$, $\angle BCD = 70^\circ$, and $BC = AD$. Find $\angle ACD.$

2014 HMMT February Geometry 5
Let $\mathcal{C}$ be a circle in the $xy$ plane with radius $1$ and center $(0, 0, 0)$, and let $P$ be a point in space with coordinates $(3, 4, 8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.

2014 HMMT February Geometry 6
In quadrilateral $ABCD$, we have $AB = 5$, $BC = 6$, $CD = 5$, $DA = 4$, and $\angle ABC = 90^\circ$. Let $AC$ and $BD$ meet at $E$. Compute $\dfrac{BE}{ED}$.

2014 HMMT February Geometry 7
Triangle $ABC$ has sides $AB = 14$, $BC = 13$, and $CA = 15$. It is inscribed in circle $\Gamma$, which has center $O$. Let $M$ be the midpoint of $AB$, let $B'$ be the point on $\Gamma$ diametrically opposite $B$, and let $X$ be the intersection of $AO$ and $MB'$. Find the length of $AX$.

2014 HMMT February Geometry 8
Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$.

2014 HMMT February Geometry 9
Two circles are said to be orthogonal if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_1$ and $\omega_2$ with radii $10$ and $13$, respectively, are externally tangent at point $P$. Another circle $\omega_3$ with radius $2\sqrt2$ passes through $P$ and is orthogonal to both $\omega_1$ and $\omega_2$. A fourth circle $\omega_4$, orthogonal to $\omega_3$, is externally tangent to $\omega_1$ and $\omega_2$. Compute the radius of $\omega_4$.

2014 HMMT February Geometry 10
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $\Gamma$ be the circumcircle of $ABC$, let $O$ be its circumcenter, and let $M$ be the midpoint of minor arc $BC$. Circle $\omega_1$ is internally tangent to $\Gamma$ at $A$, and circle $\omega_2$, centered at $M$, is externally tangent to $\omega_1$ at a point $T$. Ray $AT$ meets segment $BC$ at point $S$, such that $BS - CS = \dfrac4{15}$. Find the radius of $\omega_2$



2015 HMMT February Geometry 1

Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown.[/asy]Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.

2015 HMMT February Geometry 2

Let $ABC$ be a triangle with orthocenter $H$; suppose $AB=13$, $BC=14$, $CA=15$. Let $G_A$ be the centroid of triangle $HBC$, and define $G_B$, $G_C$ similarly. Determine the area of triangle $G_AG_BG_C$.

2015 HMMT February Geometry 3

Let $ABCD$ be a quadrilateral with $\angle BAD = \angle ABC = 90^{\circ}$, and suppose $AB=BC=1$, $AD=2$. The circumcircle of $ABC$ meets $\overline{AD}$ and $\overline{BD}$ at point $E$ and $F$, respectively. If lines $AF$ and $CD$ meet at $K$, compute $EK$.

2015 HMMT February Geometry 4

Let $ABCD$ be a cyclic quadrilateral with $AB=3$, $BC=2$, $CD=2$, $DA=4$. Let lines perpendicular to $\overline{BC}$ from $B$ and $C$ meet $\overline{AD}$ at $B'$ and $C'$, respectively. Let lines perpendicular to $\overline{BC}$ from $A$ and $D$ meet $\overline{AD}$ at $A'$ and $D'$, respectively. Compute the ratio $\frac{[BCC'B']}{[DAA'D']}$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$.

2015 HMMT February Geometry 5

Let $I$ be the set of points $(x,y)$ in the Cartesian plane such that $$x>\left(\frac{y^4}{9}+2015\right)^{1/4}$$ Let $f(r)$ denote the area of the intersection of $I$ and the disk $x^2+y^2\le r^2$ of radius $r>0$ centered at the origin $(0,0)$. Determine the minimum possible real number $L$ such that $f(r)<Lr^2$ for all $r>0$.

2015 HMMT February Geometry 6

In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.

2015 HMMT February Geometry 7

Let $ABCD$ be a square pyramid of height $\frac{1}{2}$ with square base $ABCD$ of side length $AB=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $ABCD$ is the center of square $ABCD$). The faces $ADE$ and $CDE$ meet at an acute angle of measure $\alpha$ (so that $0^{\circ}<\alpha<90^{\circ}$). Find $\tan \alpha$.

2015 HMMT February Geometry 8

Let $S$ be the set of discs $D$ contained completely in the set $\{ (x,y) : y<0\}$ (the region below the $x$-axis) and centered (at some point) on the curve $y=x^2-\frac{3}{4}$. What is the area of the union of the elements of $S$?

2015 HMMT February Geometry 9

Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in the triangle $BCD$ such that $[XBC]=2[XBD]=4[XCD]$, where $[\overline{\omega}]$ denotes the area of figure $\overline{\omega}$. Let $Y$ lie on segment $AX$ such that $2AY=YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.

2015 HMMT February Geometry 10

Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of negative slope tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a unique point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.

2016 HMMT February Geometry 1
Dodecagon $QWARTZSPHINX$ has all side lengths equal to $2$, is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$. What are all possible values of the area of $\triangle SIX$?

2016 HMMT February Geometry 2
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the distance between the circumcenters of triangles $AHB$ and $AHC$.

2016 HMMT February Geometry 3
In the below picture, $T$ is an equilateral triangle with a side length of $5$ and $\omega$ is a circle with a radius of $2$. The triangle and the circle have the same center. Let $X$ be the area of the shaded region, and let $Y$ be the area of the starred region. What is $X - Y$?

2016 HMMT February Geometry 4
Let $ABC$ be a triangle with $AB = 3$, $AC = 8$, $BC = 7$ and let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Point $T$ is selected on side $BC$ so that $AT = TC$. The circumcircles of triangles $BAT$, $MAN$ intersect at $D$. Compute $DC$.

2016 HMMT February Geometry 5
Nine pairwise noncongruent circles are drawn in the plane such that any two circles intersect twice. For each pair of circles, we draw the line through these two points, for a total of $\binom 92 = 36$ lines. Assume that all $36$ lines drawn are distinct. What is the maximum possible number of points which lie on at least two of the drawn lines?

2016 HMMT February Geometry 6
Let $ABC$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M$, $N$, $P$ be the midpoints of sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ and let $E$, $F$ be the tangency points of $\gamma$ with $\overline{CA}$ and $\overline{AB}$, respectively. Let $U$, $V$ be the intersections of line $EF$ with line $MN$ and line $MP$, respectively, and let $X$ be the midpoint of arc $\widehat{BAC}$ of $\Gamma$.
Given that $AB = 5$, $AC = 8$, and $\angle A = 60^{\circ}$, compute the area of triangle $XUV$.

2016 HMMT February Geometry 7
For $i=0,1,\dots,5$ let $l_i$ be the ray on the Cartesian plane starting at the origin, an angle $\theta=i\frac{\pi}{3}$ counterclockwise from the positive $x$-axis. For each $i$, point $P_i$ is chosen uniformly at random from the intersection of $l_i$ with the unit disk. Consider the convex hull of the points $P_i$, which will (with probability 1) be a convex polygon with $n$ vertices for some $n$. What is the expected value of $n$?

2016 HMMT February Geometry 8
In cyclic quadrilateral $ABCD$ with $AB = AD = 49$ and $AC = 73$, let $I$ and $J$ denote the incenters of triangles $ABD$ and $CBD$. If diagonal $\overline{BD}$ bisects $\overline{IJ}$, find the length of $IJ$.

2016 HMMT February Geometry 9
The incircle of a triangle $ABC$ is tangent to $BC$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$. The \emph{$B$-mixtilinear incircle}, centered at $O_B$, is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$. The \emph{$C$-mixtilinear incircle}, centered at $O_C$, is defined similarly. Suppose that $\overline{DH} \perp \overline{O_BO_C}$, $AB = \sqrt3$ and $AC = 2$. Find $BC$.

2016 HMMT February Geometry 10
The incircle of a triangle $ABC$ is tangent to $BC$ at $D$. Let $H$ and $\Gamma$ denote the orthocenter and circumcircle of $\triangle ABC$. The $B$-mixtilinear incircle, centered at $O_B$,
is tangent to lines $BA$ and $BC$ and internally tangent to $\Gamma$. The $C$-mixtilinear incircle, centered at $O_C$, is defined similarly. Suppose that $\overline{DH} \perp \overline{O_BO_C}$, $AB = \sqrt3$ and $AC = 2$. Find $BC$.




2017 HMMT February Geometry 1
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 $\frac{AP}{CP}$.

2017 HMMT February Geometry 2
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

2017 HMMT February Geometry 3
Let $S$ be a set of $2017$ points in the plane. Let $R$ be the radius of the smallest circle containing all points in $S$ on either the interior or boundary. Also, let $D$ be the longest distance between two of the points in $S$. Let $a$, $b$ be real numbers such that $a\le \frac{D}{R}\le b$ for all possible sets $S$, where $a$ is as large as possible and $b$ is as small as possible. Find the pair $(a, b)$.

2017 HMMT February Geometry 4
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

2017 HMMT February Geometry 5
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$.

2017 HMMT February Geometry 6
In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$.

2017 HMMT February Geometry 7
Let $\omega$ and $\Gamma$ be circles such that $\omega$ is internally tangent to $\Gamma$ at a point $P$. Let $AB$ be a chord of $\Gamma$ tangent to $\omega$ at a point $Q$. Let $R\neq P$ be the second intersection of line $PQ$ with $\Gamma$. If the radius of $\Gamma$ is $17$, the radius of $\omega$ is $7$, and $\frac{AQ}{BQ}=3$, find the circumradius of triangle $AQR$.

2017 HMMT February Geometry 8
Let $ABC$ be a triangle with circumradius $R=17$ and inradius $r=7$. Find the maximum possible value of $\sin \frac{A}{2}$.

2017 HMMT February Geometry 9
Let $ABC$ be a triangle, and let $BCDE$, $CAFG$, $ABHI$ be squares that do not overlap the triangle with centers $X$, $Y$, $Z$ respectively. Given that $AX=6$, $BY=7$, and $CA=8$, find the area of triangle $XYZ$.

2017 HMMT February Geometry 10
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.



2018 HMMT February Geometry 1
Triangle $GRT$ has $GR=5,$ $RT=12,$ and $GT=13.$ The perpendicular bisector of $GT$ intersects the extension of $GR$ at $O.$ Find $TO.$

2018 HMMT February Geometry 2
Points $A,B,C,D$ are chosen in the plane such that segments $AB,BC,CD,DA$ have lengths $2,7,5,12,$ respectively. Let $m$ be the minimum possible value of the length of segment $AC$ and let $M$ be the maximum possible value of the length of segment $AC.$ What is the ordered pair $(m,M)$?

2018 HMMT February Geometry 3
How many noncongruent triangles are there with one side of length $20,$ one side of length $17,$ and one $60^{\circ}$ angle?

2018 HMMT February Geometry 4
A paper equilateral triangle of side length $2$ on a table has vertices labeled $A,B,C.$ Let $M$ be the point on the sheet of paper halfway between $A$ and $C.$ Over time, point $M$ is lifted upwards, folding the triangle along segment $BM,$ while $A,B,$ and $C$ on the table. This continues until $A$ and $C$ touch. Find the maximum volume of tetrahedron $ABCM$ at any time during this process.

2018 HMMT February Geometry 5
In the quadrilateral $MARE$ inscribed in a unit circle $\omega,$ $AM$ is a diameter of $\omega,$ and $E$ lies on the angle bisector of $\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$

2018 HMMT February Geometry 6
Let $ABC$ be an equilateral triangle of side length $1.$ For a real number $0<x<0.5,$ let $A_1$ and $A_2$ be the points on side $BC$ such that $A_1B=A_2C=x,$ and let $T_A=\triangle AA_1A_2.$ Construct triangles $T_B=\triangle BB_1B_2$ and $T_C=\triangle CC_1C_2$ similarly.
There exist positive rational numbers $b,c$ such that the region of points inside all three triangles $T_A,T_B,T_C$ is a hexagon with area $$\dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}.$$ Find $(b,c).$

2018 HMMT February Geometry 7
Triangle $ABC$ has sidelengths $AB=14,AC=13,$ and $BC=15.$ Point $D$ is chosen in the interior of $\overline{AB}$ and point $E$ is selected uniformly at random from $\overline{AD}.$ Point $F$ is then defined to be the intersection point of the perpendicular to $\overline{AB}$ at $E$ and the union of segments $\overline{AC}$ and $\overline{BC}.$ Suppose that $D$ is chosen such that the expected value of the length of $\overline{EF}$ is maximized. Find $AD.$

2018 HMMT February Geometry 8
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$

2018 HMMT February Geometry 9
Po picks $100$ points $P_1,P_2,\cdots, P_{100}$ on a circle independently and uniformly at random. He then draws the line segments connecting $P_1P_2,P_2P_3,\ldots,P_{100}P_1.$ Find the expected number of regions that have all sides bounded by straight lines.

2018 HMMT February Geometry 10

Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the intersection of the circumcircle of $BCY$ with line $AB.$ Given that $Y$ is on the interior of segment $CZ$ and $YZ=3CY,$ compute $AK.$



2019 HMMT February Geometry 1
Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$?

2019 HMMT February Geometry 2
In rectangle $ABCD$, points $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that both $AF$ and $CE$ are perpendicular to diagonal $BD$. Given that $BF$ and $DE$ separate $ABCD$ into three polygons with equal area, and that $EF = 1$, find the length of $BD$.

2019 HMMT February Geometry 3
Let $AB$ be a line segment with length 2, and $S$ be the set of points $P$ on the plane such that there exists point $X$ on segment $AB$ with $AX = 2PX$. Find the area of $S$.

2019 HMMT February Geometry 4
Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

2019 HMMT February Geometry 5
Isosceles triangle $ABC$ with $AB = AC$ is inscibed is a unit circle $\Omega$ with center $O$. Point $D$ is the reflection of $C$ across $AB$. Given that $DO = \sqrt{3}$, find the area of triangle $ABC$.

2019 HMMT February Geometry 6
Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.

2019 HMMT February Geometry 7
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.

2019 HMMT February Geometry 8
In triangle $ABC$ with $AB < AC$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $OH$ lies on $BC$, $BC = 1$, and the perimeter of $ABC$ is 6, find the area of $ABC$.

2019 HMMT February Geometry 9
In a rectangular box $ABCDEFGH$ with edge lengths $AB = AD = 6$ and $AE = 49$, a plane slices through point $A$ and intersects edges $BF$, $FG$, $GH$, $HD$ at points $P$, $Q$, $R$, $S$ respectively. Given that $AP = AS$ and $PQ = QR = RS$, find the area of pentagon $APQRS$.

2019 HMMT February Geometry 10

In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon $A_4A_3B_4B_3C_4C_3$. Find the area of the hexagon $A_5A_6B_5B_6C_5C_6$.



2020 HMMT February Geometry 1
Let $DIAL$, $FOR$, and $FRIEND$ be regular polygons in the plane. If $ID=1$, find the product of all possible areas of $OLA$.

by Andrew Gu

2020 HMMT February Geometry 2
Let $ABC$ be a triangle with $AB=5$, $AC=8$, and $\angle BAC=60^\circ$. Let $UVWXYZ$ be a regular hexagon that is inscribed inside $ABC$ such that $U$ and $V$ lie on side $BA$, $W$ and $X$ lie on side $AC$, and $Z$ lies on side $CB$. What is the side length of hexagon $UVWXYZ$?

by Ryan Kim

2020 HMMT February Geometry 3
Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?

by James Lin

2020 HMMT February Geometry 4
Let $ABCD$ be a rectangle and $E$ be a point on segment $AD$. We are given that quadrilateral $BCDE$ has an inscribed circle $\omega_1$ that is tangent to $BE$ at $T$. If the incircle $\omega_2$ of $ABE$ is also tangent to $BE$ at $T$, then find the ratio of the radius of $\omega_1$ to the radius of $\omega_2$.

by James Lin

2020 HMMT February Geometry 5
Let $ABCDEF$ be a regular hexagon with side length $2$. A circle with radius $3$ and center at $A$ is drawn. Find the area inside quadrilateral $BCDE$ but outside the circle.

by Carl Joshua Quines

2020 HMMT February Geometry 6
Let $ABC$ be a triangle with $AB=5$, $BC=6$, $CA=7$. Let $D$ be a point on ray $AB$ beyond $B$ such that $BD=7$, $E$ be a point on ray $BC$ beyond $C$ such that $CE=5$, and $F$ be a point on ray $CA$ beyond $A$ such that $AF=6$. Compute the area of the circumcircle of $DEF$.

by James Lin

2020 HMMT February Geometry 7

Let $\Gamma$ be a circle, and $\omega_1$ and $\omega_2$ be two non-intersecting circles inside $\Gamma$ that are internally tangent to $\Gamma$ at $X_1$ and $X_2$, respectively. Let one of the common internal tangents of $\omega_1$ and $\omega_2$ touch $\omega_1$ and $\omega_2$ at $T_1$ and $T_2$, respectively, while intersecting $\Gamma$ at two points $A$ and $B$. Given that $2X_1T_1=X_2T_2$ and that $\omega_1$, $\omega_2$, and $\Gamma$ have radii $2$, $3$, and $12$, respectively, compute the length of $AB$.

by James Lin

2020 HMMT February Geometry 8
Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$, respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$. Given that $EV=6$, $VA'=7$, and $A'N=9$, compute the radius of $\Gamma$.

by James Lin

2020 HMMT February Geometry 9
Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$.

by Andrew Gu

2020 HMMT February Geometry 10

Let $\Gamma$ be a circle of radius $1$ centered at $O$. A circle $\Omega$ is said to be \emph{friendly} if there exist distinct circles $\omega_1$, $\omega_2$, $\ldots$, $\omega_{2020}$, such that for all $1\le i\le2020$, $\omega_i$ is tangent to $\Gamma$, $\Omega$, and $\omega_{i+1}$. (Here, $\omega_{2021} = \omega_1$.) For each point $P$ in the plane, let $f(P)$ denote the sum of the areas of all friendly circles centered at $P$. If $A$ and $B$ are points such that $OA=\frac12$ and $OB=\frac13$, determine $f(A)-f(B)$.

by Michael Ren