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US Math Competition Association 2019-21 (USMCA) 26p

geometry problems from US Math Competition Association (USMCA) 
with aops links in the names


 aops posts collection
2019 USMCA , 2020 USMCA , 2021 USMCA

2019


2019 USMCA Challenger Division p7
Triangle $ABC$ has $AB = 8, AC = 12, BC = 10$. Let $D$ be the intersection of the angle bisector of angle $A$ with $BC$. Let $M$ be the midpoint of $BC$. The line parallel to $AC$ passing through $M$ intersects $AB$ at $N$. The line parallel to $AB$ passing through $D$ intersects $AC$ at $P$. $MN$ and $DP$ intersect at $E$. Find the area of $ANEP$.

 2019 USMCA Challenger Division p11
Let $ABC$ be a right triangle with hypotenuse $AB$. Point $E$ is on $AB$ with $AE = 10BE$, and point $D$ is outside triangle $ABC$ such that $DC = DB$ and $\angle CDA = \angle BDE$. Let $[ABC]$ and $[BCD]$ denote the areas of triangles $ABC$ and $BCD$. Determine the value of $\frac{[BCD]}{[ABC]}$.

 2019 USMCA Challenger Division p14
In a circle of radius $10$, three congruent chords bound an equilateral triangle with side length $8$. The endpoints of these chords form a convex hexagon. Compute the area of this hexagon.

 2019 USMCA Challenger Division p18
Two circles with radii $3$ and $4$ are externally tangent at $P$. Let $A \neq P$ be on the first circle and $B \neq P$ be on the second circle, and let the tangents at $A$ and $B$ to the respective circles intersect at $Q$. Given that $QA = QB$ and $AB$ bisects $PQ$, compute the area of $QAB$.

 2019 USMCA Challenger Division p20
Kelvin the Frog lives in the 2-D plane. Each day, he picks a uniformly random direction (i.e. a uniformly random bearing $\theta\in [0,2\pi]$) and jumps a mile in that direction. Let $D$ be the number of miles Kelvin is away from his starting point after ten days. Determine the expected value of $D^4$.

 2019 USMCA Challenger Division p21
Let $ABCD$ be a rectangle satisfying $AB = CD = 24$, and let $E$ and $G$ be points on the extension of $BA$ past $A$ and the extension of $CD$ past $D$ respectively such that $AE = 1$ and $DG = 3$.
Suppose that there exists a unique pair of points $(F, H)$ on lines $BC$ and $DA$ respectively such that $H$ is the orthocenter of $\triangle EFG$. Find the sum of all possible values of $BC$.

 2019 USMCA Challenger Division p24
Let $ABC$ be a triangle with $\angle A = 60^\circ$, $AB = 12$, $AC = 14$. Point $D$ is on $BC$ such that $\angle BAD = \angle CAD$. Extend $AD$ to meet the circumcircle at $M$. The circumcircle of $BDM$ intersects $AB$ at $K \neq B$, and line $KM$ intersects the circumcircle of $CDM$ at $L \neq M$. Find $\frac{KM}{LM}$.

 2019 USMCA Challenger Division p30
Let $ABC$ be a triangle with $BC = a$, $CA = b$, and $AB = c$. The $A$-excircle is tangent to $\overline{BC}$ at $A_1$; points $B_1$ and $C_1$ are similarly defined.
Determine the number of ways to select positive integers $a$, $b$, $c$ such that the numbers $-a+b+c$, $a-b+c$, and $a+b-c$ are even integers at most 100, and the circle through the midpoints of $\overline{AA_1}$, $\overline{BB_1}$, and $\overline{CC_1}$ is tangent to the incircle of $\triangle ABC$.
2019 USMCA Premier Division p3
Let $ABC$ be a scalene triangle. The incircle of $ABC$ touches $\overline{BC}$ at $D$. Let $P$ be a point on $\overline{BC}$ satisfying $\angle BAP = \angle CAP$, and $M$ be the midpoint of $\overline{BC}$. Define $Q$ to be on $\overline{AM}$ such that $\overline{PQ} \perp \overline{AM}$. Prove that the circumcircle of $\triangle AQD$ is tangent to $\overline{BC}$.

 2019 USMCA Premier Division p7
Let $AXBY$ be a convex quadrilateral. The incircle of $\triangle AXY$ has center $I_A$ and touches $\overline{AX}$ and $\overline{AY}$ at $A_1$ and $A_2$ respectively. The incircle of $\triangle BXY$ has center $I_B$ and touches $\overline{BX}$ and $\overline{BY}$ at $B_1$ and $B_2$ respectively. Define $P = \overline{XI_A} \cap \overline{YI_B}$, $Q = \overline{XI_B} \cap \overline{YI_A}$, and $R = \overline{A_1B_1} \cap \overline{A_2B_2}$.
Prove that if $\angle AXB = \angle AYB$, then $P$, $Q$, $R$ are collinear.
Prove that if there exists a circle tangent to all four sides of $AXBY$, then $P$, $Q$, $R$ are collinear.

2020 

2020 USMCA Online Qualifier p5
A unit square $ABCD$ is balanced on a flat table with only its vertex $A$ touching the table, such that $AC$ is perpendicular to the table. The square loses balance and falls to one side. At the end of the fall, $A$ is in the same place as before, and $B$ is also touching the table. Compute the area swept by the square during its fall.


2020 USMCA Online Qualifier p8
Two right cones each have base radius 4 and height 3, such that the apex of each cone is the center of the base of the other cone. Find the surface area of the union of the cones.

2020 USMCA Online Qualifier p10
Let $ABCD$ be a unit square, and let $E$ be a point on segment $AC$ such that $AE = 1$. Let $DE$ meet $AB$ at $F$ and $BE$ meet $AD$ at $G$. Find the area of $CFG$.

2020 USMCA Online Qualifier p13
$\Omega$ is a quarter-circle of radius $1$. Let $O$ be the center of $\Omega$, and $A$ and $B$ be the endpoints of its arc. Circle $\omega$ is inscribed in $\Omega$. Circle $\gamma$ is externally tangent to $\omega$ and internally tangent to $\Omega$ on segment $OA$ and arc $AB$. Determine the radius of $\gamma$.

2020 USMCA Online Qualifier p20
Let $\Omega$ be a circle centered at $O$. Let $ABCD$ be a quadrilateral inscribed in $\Omega$, such that $AB = 12$, $AD = 18$, and $AC$ is perpendicular to $BD$. The circumcircle of $AOC$ intersects ray $DB$ past $B$ at $P$. Given that $\angle PAD = 90^\circ$, find $BD^2$.

2020 USMCA Online Qualifier p25
Let $AB$ be a segment of length $2$. The locus of points $P$ such that the $P$-median of triangle $ABP$ and its reflection over the $P$-angle bisector of triangle $ABP$ are perpendicular determines some region $R$. Find the area of $R$.


Let $ABCDEF$ be a regular hexagon with side length two. Extend $FE$ and $BD$ to meet at $G$. Compute the area of $ABGF$.

Two altitudes of a triangle have lengths $8$ and $15$. How many possible integer lengths are there for the third altitude?

Let $\Omega$ be a unit circle and $A$ be a point on $\Omega$. An angle $0 < \theta < 180^\circ$ is chosen uniformly at random, and $\Omega$ is rotated $\theta$ degrees clockwise about $A$. What is the expected area swept by this rotation?

Equiangular octagon $ABCDEFGH$ is inscribed in a circle centered at $O$. Chords $AD$ and $BG$ intersect at $K$. Given that $AB = 2$ and the octagon has area $15$, compute the area of $HAKBO$.

Triangle $ABC$ has $BC = 7, CA = 8, AB = 9$. Let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively, and let $G$ be the intersection of $AD$ and $BE$. $G'$ is the reflection of $G$ across $D$. Let $G'E$ meet $CG$ at $P$, and let $G'F$ meet $BG$ at $Q$. Determine the area of $APG'Q$.

Let $ABCDEF$ be a regular octahedron with unit side length, such that $ABCD$ is a square. Points $G, H$ are on segments $BE, DF$ respectively. The planes $AGD$ and $BCH$ divide the octahedron into three pieces, each with equal volume. Compute $BG$.

Let $\Gamma$ be a circle centered at $O$ with chord $AB$. The tangents to $\Gamma$ at $A$ and $B$ meet at $C$. A secant from $C$ intersects chord $AB$ at $D$ and $\Gamma$ at $E$ such that $D$ lies on segment $CE$. Given that $\angle BOD + \angle EAD = 180^\circ$, $AE = 1$, and $BE = 2$, find $CE$.

Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$.


Let $ABC$ be an acute triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Show that $BCHG$ is a parallelogram.

Let $ABCD$ be a convex quadrilateral, and let $\omega_A$ and $\omega_B$ be the incircles of $\triangle ACD$ and $\triangle BCD$, with centers $I$ and $J$. The second common external tangent to $\omega_A$ and $\omega_B$ touches $\omega_A$ at $K$ and $\omega_B$ at $L$. Prove that lines $AK$, $BL$, $IJ$ are concurrent.

                                                       2021 

Let $ABCD$ be a unit square. Construct point $E$ outside $ABCD$ such that $\overline{AE} = \sqrt{2} \cdot \overline{BE}$ and $\angle{AEB} = 135^{\circ}$. Also, let $F$ be the foot of the perpendicular from $A$ to line $BE$. Find the area of $\triangle{BDF}$.

Let $ABCD$ be a parallelogram with $AB=CD=16$ and $BC=AD=24.$ Suppose the angle bisectors of $\angle A$ and $\angle D$ intersect $BC$ at $E$ and $F,$ respectively. Moreover, suppose $AE$ and $DF$ intersect at $P.$ Given that the sum of the areas of quadrilaterals $ABFP$ and $DCEP$ is $100,$ compute the area of the parallelogram.

Let $\mathcal{C}$ be a right circular cone with height $\sqrt{15}$ and base radius $1$. Let $V$ be the vertex of $\mathcal{C}$, $B$ be a point on the circumference of the base of $\mathcal{C}$, and $A$ be the midpoint of $VB$. An ant travels at constant velocity on the surface of the cone from $A$ to $B$ and makes two complete revolutions around $\mathcal{C}$. Find the distance the ant travelled.

Let $X_1X_2X_3X_4$ be a quadrilateral inscribed in circle $\Omega$ such that $\triangle{X_1X_2X_3}$ has side lengths $13,14,15$ in some order. For $1 \le i \le 4$, let $l_i$ denote the tangent to $\Omega$ at $X_i$, and let $Y_i$ denote the intersection of $l_i$ and $l_{i+1}$ (indices taken modulo $4$). Find the least possible area of $Y_1Y_2Y_3Y_4$.

Let $ABC$ be an equilateral triangle with unit side length and circumcircle $\Gamma$. Let $D_1, D_2$ be the points on $\Gamma$ such that $BD_i = 3CD_i$. Let $E_1, E_2$ be the points on $\Gamma$ such that $CE_i = 3AE_i$. Let $F_1, F_2$ be the points on $\Gamma$ such that $AF_i = 3BF_i$. Then points $D_1, D_2, E_1, E_2, F_1, F_2$ are the vertices of a convex hexagon. What is the area of this hexagon?

Let $ABC$ be a triangle with $AB=20, AC=21,$ and $\angle BAC = 90^{\circ}.$ Suppose $\Gamma_1$ is the unique circle centered at $B$ and passing through $A,$ and $\Gamma_2$ is the unique circle centered at $C$ and passing through $A.$ Points $E$ and $F$ are selected on $\Gamma_1$ and $\Gamma_2,$ respectively, such that $E, A, F$ are collinear in that order. The tangent to $\Gamma_1$ at $E$ and the tangent to $\Gamma_2$ at $F$ intersect at $P$. Given that $PA \bot BC$, compute the area of $PBC$.

Convex equiangular hexagon $ABCDEF$ has $AB = CD = EF = \sqrt 3$ and $BC = DE = FA = 2.$ Points $X, Y,$ and $Z$ are situated outside the hexagon such that $AEX, ECY,$ and $CAZ$ are all equilateral triangles. Compute the area of the region bounded by lines $XF, YD, $ and $ZB.$

Three circles $\Gamma_A, \Gamma_B, \Gamma_C$ are externally tangent. The circles are centered at $A, B, C$ and have radii $4, 5, 6$ respectively. Circles $\Gamma_B$ and $\Gamma_C$ meet at $D$, circles $\Gamma_C$ and $\Gamma_A$ meet at $E$, and circles $\Gamma_A$ and $\Gamma_B$ meet at $F$. Let $GH$ be a common external tangent of $\Gamma_B$ and $\Gamma_C$ on the opposite side of $BC$ as $EF$, with $G$ on $\Gamma_B$ and $H$ on $\Gamma_C$. Lines $FG$ and $EH$ meet at $K$. Point $L$ is on $\Gamma_A$ such that $\angle DLK = 90^\circ$. Compute $\frac{LG}{LH}$.




source: www.usmath.org

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