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US Math Competition Association 2019-20 (USMCA) 17p

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

aops posts collection
2019 USMCA , 2020 USMCA Online Qualifier


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 online qualifier

2019 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.

2019 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.

2019 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 $CFG$.

2019 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 $\gamma$.

2019 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 $BD^2$.

2019 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 $R$.

source: www.usmath.org

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