[b]p1.[/b] David runs at $3$ times the speed of Alice. If Alice runs $2$ miles in $30$ minutes, determine how many minutes it takes for David to run a mile.


[b]p2.[/b] Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$.


[b]p3.[/b] Find the $7$-digit palindrome which is divisible by $7$ and whose first three digits are all $2$.


[b]p4.[/b] Determine the number of ways to put $5$ indistinguishable balls in $6$ distinguishable boxes.


[b]p5.[/b] A certain reduced fraction $\frac{a}{b}$ (with $a,b > 1$) has the property that when $2$ is subtracted from the numerator and added to the denominator, the resulting fraction has $\frac16$ of its original value. Find this fraction.


[b]p6.[/b] Find the smallest positive integer $n$ such that $|\tau(n +1)-\tau(n)| = 7$. Here, $\tau(n)$ denotes the number of divisors of $n$.


[b]p7.[/b] Let $\vartriangle ABC$ be the triangle such that $AB = 3$, $AC = 6$ and $\angle BAC = 120^o$. Let $D$ be the point on $BC$ such that $AD$ bisect $\angle BAC$. Compute the length of $AD$.


[b]p8.[/b] $26$ points are evenly spaced around a circle and are labeled $A$ through $Z$ in alphabetical order. Triangle $\vartriangle LMT$ is drawn. Three more points, each distinct from $L, M$, and $T$ , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap.


[b]p9.[/b] Given the three equations
$a +b +c = 0$
$a^2 +b^2 +c^2 = 2$
$a^3 +b^3 +c^3 = 19$
find $abc$.


[b]p10.[/b] Circle $\omega$ is inscribed in convex quadrilateral $ABCD$ and tangent to $AB$ and $CD$ at $P$ and $Q$, respectively. Given that $AP = 175$, $BP = 147$,$CQ = 75$, and $AB \parallel CD$, find the length of $DQ$.


[b]p11. [/b]Let $p$ be a prime and m be a positive integer such that $157p = m^4 +2m^3 +m^2 +3$. Find the ordered pair $(p,m)$.


[b]p12.[/b] Find the number of possible functions $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$ that satisfy the following conditions.
(1) $f (x) \ne f (y)$ when $x \ne y$ 
(2) There exists some $x$ such that $f (x)^2 = x^2$


[b]p13.[/b] Let $p$ be a prime number such that there exists positive integer $n$ such that $41pn -42p^2 = n^3$. Find the sum of all possible values of $p$.


[b]p14.[/b] An equilateral triangle with side length $ 1$ is rotated $60$ degrees around its center. Compute the area of the region swept out by the interior of the triangle.


[b]p15.[/b] Let $\sigma (n)$ denote the number of positive integer divisors of $n$. Find the sum of all $n$ that satisfy the equation $\sigma (n) =\frac{n}{3}$.


[b]p16[/b]. Let $C$ be the set of points $\{a,b,c\} \in Z$ for $0  \le a,b,c \le 10$. Alice starts at $(0,0,0)$. Every second she randomly moves to one of the other points in $C$ that is on one of the lines parallel to the $x, y$, and $z$ axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach $(10,10,10)$.


[b]p17.[/b] Find the maximum possible value of $$abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3$$ where $a,b,c$ are real such that $a +b +c = 0$.


[b]p18.[/b] Circle $\omega$ with radius $6$ is inscribed within quadrilateral $ABCD$. $\omega$ is tangent to $AB$, $BC$, $CD$, and $DA$ at $E, F, G$, and $H$ respectively. If $AE = 3$, $BF = 4$ and $CG = 5$, find the length of $DH$.


[b]p19.[/b] Find the maximum integer $p$ less than $1000$ for which there exists a positive integer $q$ such that the cubic equation $$x^3 - px^2 + q x -(p^2 -4q +4) = 0$$ has three roots which are all positive integers.


[b]p20.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle ABC = 60^o$,$\angle ACB = 20^o$. Let $P$ be the point such that $CP$ bisects $\angle ACB$ and $\angle PAC = 30^o$. Find $\angle PBC$.


PS. You had better use hide for answers.
To solve the problem, we need to find the maximum possible value of the expression 
\[ abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)^3 \]
where \(a, b, c\) are real numbers such that \(a + b + c = 0\).

1. First, we rewrite the given expression:
\[
abc \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)^3 = abc \left( \frac{ab + ac + bc}{abc} \right)^3 = \frac{(ab + ac + bc)^3}{(abc)^2}
\]

2. Next, we use the condition \(a + b + c = 0\) to express \(ab + ac + bc\) in terms of \(a, b, c\). Squaring both sides of \(a + b + c = 0\), we get:
\[
(a + b + c)^2 = 0 \implies a^2 + b^2 + c^2 + 2(ab + ac + bc) = 0 \implies ab + ac + bc = -\frac{a^2 + b^2 + c^2}{2}
\]

3. Substitute \(ab + ac + bc = -\frac{a^2 + b^2 + c^2}{2}\) into the expression:
\[
\frac{(ab + ac + bc)^3}{(abc)^2} = \frac{\left( -\frac{a^2 + b^2 + c^2}{2} \right)^3}{(abc)^2} = \frac{-(a^2 + b^2 + c^2)^3}{8(abc)^2}
\]

4. To find the maximum value, we need to consider the constraints on \(a, b, c\). By the Arithmetic Mean-Geometric Mean (AM-GM) inequality, we know:
\[
a^2 + b^2 + c^2 \geq 3\sqrt[3]{a^2b^2c^2}
\]

5. Since \(a + b + c = 0\), we can use the fact that \(a, b, c\) are not all zero. We can assume without loss of generality that \(a = x, b = y, c = -x - y\). Then:
\[
a^2 + b^2 + c^2 = x^2 + y^2 + (-x - y)^2 = x^2 + y^2 + x^2 + 2xy + y^2 = 2x^2 + 2y^2 + 2xy
\]

6. Substitute \(a = x, b = y, c = -x - y\) into the expression:
\[
abc = x y (-x - y) = -x^2 y - xy^2
\]

7. To maximize the expression, we need to find the values of \(x\) and \(y\) that satisfy the conditions. By symmetry and trial, we find that the maximum value occurs when \(a = b = -c\). Let \(a = b = 1\) and \(c = -2\):
\[
abc = 1 \cdot 1 \cdot (-2) = -2
\]
\[
ab + ac + bc = 1 \cdot 1 + 1 \cdot (-2) + 1 \cdot (-2) = 1 - 2 - 2 = -3
\]
\[
\frac{(ab + ac + bc)^3}{(abc)^2} = \frac{(-3)^3}{(-2)^2} = \frac{-27}{4} = -6.75
\]

8. However, we need to find the maximum positive value. By further analysis, we find that the maximum value is achieved when \(a = b = c = 0\), which is not possible. Therefore, we use the AM-GM inequality to find the maximum value:
\[
\frac{8(ab + bc + ac)^3}{27} \le (abc)^2
\]

9. The maximum value is achieved when \(a = b = c\), giving:
\[
\boxed{\frac{27}{8}}
\]