[u]Round 9[/u]


[b]p25.[/b] Circle $\omega_1$ has radius $1$ and diameter $AB$. Let circle $\omega_2$ be a circle withm aximum radius such that it is tangent to $AB$ and internally tangent to $\omega_1$. A point $C$ is then chosen such that $\omega_2$ is the incircle of triangle $ABC$. Compute the area of $ABC$.


[b]p26.[/b] Two particles lie at the origin of a Cartesian plane. Every second, the first particle moves from its initial position $(x, y)$ to one of either $(x +1, y +2)$ or $(x -1, y -2)$, each with probability $0.5$. Likewise, every second the second particle moves from it’s initial position $(x, y)$ to one of either $(x +2, y +3)$ or $(x -2, y -3)$, each with probability $0.5$. Let $d$ be the distance distance between the two particles after exactly one minute has elapsed. Find the expected value of $d^2$.


[b]p27.[/b] Find the largest possible positive integer $n$ such that for all positive integers $k$ with $gcd (k,n) = 1$, $k^2 -1$ is a multiple of $n$.


[u]Round 10[/u]


[b]p28.[/b]  Let $\vartriangle ABC$ be a triangle with side lengths $AB = 13$, $BC = 14$, $C A = 15$. Let $H$ be the orthcenter of  $\vartriangle ABC$, $M$ be the midpoint of segment $BC$, and $F$ be the foot of altitude from $C$ to $AB$. Let $K$ be the point on line $BC$ such that $\angle MHK = 90^o$. Let $P$ be the intersection of $HK$ and $AB$. Let $Q$ be the intersection of circumcircle of $\vartriangle FPK$ and $BC$. Find the length of $QK$.


[b]p29.[/b] Real numbers $(x, y, z)$ are chosen uniformly at random from the interval $[0,2\pi]$. Find the probability that $$\cos (x) \cdot \cos (y)+ \cos(y) \cdot \cos (z)+ \cos (z) \cdot \cos(x) + \sin (x) \cdot  \sin (y)+ \sin (y) \cdot  \sin (z)+ \sin (z) \cdot \sin (x)+1$$ is positive.


[b]p30.[/b] Find the number of positive integers where each digit is either $1$, $3$, or $4$, and the sum of the digits is $22$.


[u]Round 11[/u]


[b]p31.[/b]  In $\vartriangle ABC$, let $D$ be the point on ray  $\overrightarrow{CB}$ such that $AB = BD$ and let $E$ be the point on ray $\overrightarrow{AC}$ such that $BC =CE$. Let $L$ be the intersection of $AD$ and circumcircle of $\vartriangle ABC$. The exterior angle bisector of $\angle C$ intersects $AD$ at $K$ and it follows that $AK = AB +BC +C A$. Given that points $B$, $E$, and $L$ are collinear, find $\angle C AB$.


[b]p32.[/b] Let $a$ be the largest root of the equation $x^3 -3x^2 +1  0$. Find the remainder when $\lfloor a^{2019} \rfloor$ is divided by $17$.


[b]p33.[/b]  For all $x, y \in Q$, functions $f , g ,h : Q \to Q$ satisfy $f (x + g (y)) = g (h( f (x)))+ y$. If $f (6)=2$, $g\left( \frac12 \right) = 2$, and $h \left( \frac72 \right)= 2$, find all possible values of $f (2019)$.


[u]Round 12[/u]  


[b]p34.[/b]  An $n$-polyomino is formed by joining $n$ unit squares along their edges. A free polyomino is a polyomino considered up to congruence. That is, two free polyominos are the same if there is a combination of translations, rotations, and reflections that turns one into the other. Let $P(n)$ be the number of free $n$-polyominos. For example, $P(3) = 2$ and $P(4) = 5$. Estimate $P(20)+P(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$


[b]p35.[/b]  Estimate $$\sum^{2019}_{k=1} sin(k),$$
where $k$ is measured in radians. If your estimate is $E$ and the actual value is $A$, your score for this problem will be $\max \, (0,15-10 \cdot |E - A|)$ .


[b]p36.[/b] For a positive integer $n$, let $r_{10}(n)$ be the number of $10$-tuples of (not necessarily positive) integers $(a_1,a_2,... ,a_9,a_{10})$ such that $$a^2_1 +a^2_2+ ...+a^2_9+a^2_{10}= n.$$ Estimate $r_{10}(20)+r_{10}(19)$. If your estimate is $E$ and the actual value is $A$, your score for this problem will be$$\max \, \left( 0, \left \lfloor 15-10 \cdot \left|\log_{10} \left( \frac{A}{E} \right) \right| \right \rfloor \right).$$


PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3165997p28809441]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166012p28809547]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
**Problem 25:** Circle $\omega_1$ has radius $1$ and diameter $AB$. Let circle $\omega_2$ be a circle with maximum radius such that it is tangent to $AB$ and internally tangent to $\omega_1$. A point $C$ is then chosen such that $\omega_2$ is the incircle of triangle $ABC$. Compute the area of $ABC$.

1. **Determine the radius of $\omega_2$:**
   - Since $\omega_1$ has a radius of $1$, its diameter $AB$ is $2$.
   - $\omega_2$ is tangent to $AB$ and internally tangent to $\omega_1$. The maximum radius of $\omega_2$ is achieved when it is tangent to $AB$ at its midpoint and internally tangent to $\omega_1$ at the same point.
   - The radius of $\omega_2$ is half the radius of $\omega_1$, which is $\frac{1}{2}$.

2. **Positioning $\omega_2$ and point $C:**
   - Place $\omega_1$ centered at the origin $(0,0)$.
   - $A$ and $B$ are at $(-1,0)$ and $(1,0)$ respectively.
   - $\omega_2$ is centered at $(0, \frac{1}{2})$ with radius $\frac{1}{2}$.
   - Point $C$ is chosen such that $\omega_2$ is the incircle of $\triangle ABC$.

3. **Determine the coordinates of $C:**
   - Since $\omega_2$ is the incircle, $C$ must be vertically above the center of $\omega_2$.
   - The height from $AB$ to $C$ is the sum of the radius of $\omega_2$ and the radius of $\omega_1$, which is $1 + \frac{1}{2} = \frac{3}{2}$.
   - Therefore, $C$ is at $(0, \frac{3}{2})$.

4. **Calculate the area of $\triangle ABC:**
   - The base $AB$ is $2$ units.
   - The height from $AB$ to $C$ is $\frac{3}{2}$ units.
   - The area of $\triangle ABC$ is given by:
     \[
     \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 2 \times \frac{3}{2} = \frac{3}{2}
     \]

The final answer is $\boxed{\frac{3}{2}}$