US Math Competition Association 2019 (USMCA) 11p

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


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.


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