[u]Round  1[/u]


[b]p1.[/b] A box contains $1$ ball labelledW, $1$ ball labelled $E$, $1$ ball labelled $L$, $1$ ball labelled $C$, $1$ ball labelled $O$, $8$ balls labelled $M$, and $1$ last ball labelled $E$. One ball is randomly drawn from the box. The probability that the ball is labelled $E$ is $\frac{1}{a}$ . Find $a$.


[b]p2.[/b] Let $$G +E +N = 7$$
$$G +E +O = 15$$
$$N +T = 22.$$
Find the value of $T +O$.


[b]p3.[/b] The area of $\vartriangle LMT$ is $22$. Given that $MT = 4$ and that there is a right angle at $M$, find the length of $LM$.


[u]Round 2[/u]


[b]p4.[/b] Kevin chooses a positive $2$-digit integer, then adds $6$ times its unit digit and subtracts $3$ times its tens digit from itself. Find the greatest common factor of all possible resulting numbers.


[b]p5.[/b] Find the maximum possible number of times circle $D$ can intersect pentagon $GRASS'$ over all possible choices of points $G$, $R$, $A$, $S$, and $S'$.


[b]p6.[/b] Find the sum of the digits of the integer solution to $(\log_2 x) \cdot (\log_4 \sqrt{x}) = 36$.


[u]Round 3[/u]


[b]p7.[/b] Given that $x$ and $y$ are positive real numbers such that $x^2 + y = 20$, the maximum possible value of $x + y$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.


[b]p8.[/b] In $\vartriangle DRK$, $DR = 13$, $DK = 14$, and $RK = 15$. Let $E$ be the point such that $ED = ER = EK$. Find the value of $\lfloor DE +RE +KE \rfloor$.


[b]p9.[/b] Subaru the frog lives on lily pad $1$. There is a line of lily pads, numbered $2$, $3$, $4$, $5$, $6$, and $7$. Every minute, Subaru jumps from his current lily pad to a lily pad whose number is either $1$ or $2$ greater, chosen at random from valid possibilities. There are alligators on lily pads $2$ and $5$. If Subaru lands on an alligator, he dies and time rewinds back to when he was on lily pad number $1$. Find how many times Subaru is expected to die before he reaches pad $7$.


[u]Round 4[/u]


[b]p10.[/b] Find the sum of the following series: $$\sum^{\infty}_{i=1} = \frac{\sum^i_{j=1} j}{2^i}=\frac{1}{2^1}+\frac{1+2}{2^2}+\frac{1+2+3}{2^3}+\frac{1+2+3+4}{2^4}+... $$


[b]p11.[/b] Let $\phi (x)$ be the number of positive integers less than or equal to $x$ that are relatively prime to $x$. Find the sum of all $x$ such that $\phi (\phi(x)) = x -3$. Note that $1$ is relatively prime to every positive integer.


[b]p12.[/b] On a piece of paper, Kevin draws a circle. Then, he draws two perpendicular lines. Finally, he draws two perpendicular rays originating from the same point (an $L$ shape). What is the maximum number of sections into which the lines and rays can split the circle? 


[u]Round 5 [/u]


[b]p13.[/b] In quadrilateral $ABCD$, $\angle A = 90^o$, $\angle C = 60^o$, $\angle ABD = 25^o$, and $\angle BDC = 5^o$. Given that $AB = 4\sqrt3$, the area of quadrilateral $ABCD$ can be written as $a\sqrt{b}$. Find $10a +b$.


[b]p14.[/b] The value of $$\sum^6_{n=2} \left( \frac{n^4 +1}{n^4 -1}\right) -2 \sum^6_{n=2}\left(\frac{n^3 -n^2+n}{n^4 -1}\right)$$   can be written as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.


[b]p15.[/b] Positive real numbers $x$ and $y$ satisfy the following $2$ equations. 
$$x^{1+x^{1+x^{1+...}}}= 8$$
$$\sqrt[24]{y +\sqrt[24]{y + \sqrt[24]{y +...}}} = x$$
Find the value of $\lfloor y \rfloor$.


PS. You should use hide for answers.  Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167130p28823260]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the maximum possible value of \( x + y \) given the equation \( x^2 + y = 20 \), we can proceed as follows:

1. Express \( y \) in terms of \( x \):
   \[
   y = 20 - x^2
   \]

2. Substitute \( y \) into the expression \( x + y \):
   \[
   x + y = x + (20 - x^2) = 20 - x^2 + x
   \]

3. To find the maximum value of \( 20 - x^2 + x \), we need to find the critical points by taking the derivative and setting it to zero:
   \[
   f(x) = 20 - x^2 + x
   \]
   \[
   f'(x) = -2x + 1
   \]
   \[
   -2x + 1 = 0
   \]
   \[
   x = \frac{1}{2}
   \]

4. Substitute \( x = \frac{1}{2} \) back into the equation for \( y \):
   \[
   y = 20 - \left( \frac{1}{2} \right)^2 = 20 - \frac{1}{4} = \frac{80}{4} - \frac{1}{4} = \frac{79}{4}
   \]

5. Calculate \( x + y \) at \( x = \frac{1}{2} \):
   \[
   x + y = \frac{1}{2} + \frac{79}{4} = \frac{2}{4} + \frac{79}{4} = \frac{81}{4}
   \]

6. The maximum value of \( x + y \) is \( \frac{81}{4} \), which can be written as \( \frac{a}{b} \) where \( a = 81 \) and \( b = 4 \). Since \( a \) and \( b \) are relatively prime, we have:
   \[
   a + b = 81 + 4 = 85
   \]

The final answer is \( \boxed{ 85 } \).