[b]p1.[/b] Two robots race on the plane from $(0, 0)$ to $(a, b)$, where $a$ and $b$ are positive real numbers with $a < b$. The robots move at the same constant speed. However, the first robot can only travel in directions parallel to the lines $x = 0$ or $y = 0$, while the second robot can only travel in directions parallel to the lines $y = x$ or $y = -x$. Both robots take the shortest possible path to $(a, b)$ and arrive at the same time. Find the ratio $\frac{a}{b}$ .


[b]p2.[/b] Suppose $x + \frac{1}{x} + y + \frac{1}{y} = 12$ and $x^2 + \frac{1}{x^2} + y^2 + \frac{1}{y^2} = 70$. Compute $x^3 + \frac{1}{x^3} + y^3 + \frac{1}{y^3}$.


[b]p3.[/b] Find the largest non-negative integer $a$ such that $2^a$ divides $$3^{2^{2018}}+ 3.$$


[b]p4.[/b] Suppose $z$ and $w$ are complex numbers, and $|z| = |w| = z \overline{w}+\overline{z}w = 1$. Find the largest possible value of $Re(z + w)$, the real part of $z + w$.


[b]p5.[/b] Two people, $A$ and $B$, are playing a game with three piles of matches. In this game, a move consists of a player taking a positive number of matches from one of the three piles such that the number remaining in the pile is equal to the nonnegative difference of the numbers of matches in the other two piles. $A$ and $B$ each take turns making moves, with $A$ making the first move. The last player able to make a move wins. Suppose that the three piles have $10$, $x$, and $30$ matches. Find the largest value of $x$ for which $A$ does not have a winning strategy.


[b]p6.[/b] Let $A_1A_2A_3A_4A_5A_6$ be a regular hexagon with side length $1$. For $n = 1$,$...$, $6$, let $B_n$ be a point on the segment $A_nA_{n+1}$ chosen at random (where indices are taken mod $6$, so $A_7 = A_1$). Find the expected area of the hexagon $B_1B_2B_3B_4B_5B_6$.


[b]p7.[/b] A termite sits at the point $(0, 0, 0)$, at the center of the octahedron $|x| + |y| + |z| \le 5$. The termite can only move a unit distance in either direction parallel to one of the $x$, $y$, or $z$ axes: each step it takes moves it to an adjacent lattice point. How many distinct paths, consisting of $5$ steps, can the termite use to reach the surface of the octahedron?


[b]p8.[/b] Let $$P(x) = x^{4037} - 3 - 8 \cdot \sum^{2018}_{n=1}3^{n-1}x^n$$
Find the number of roots $z$ of $P(x)$ with $|z| > 1$, counting multiplicity.


[b]p9.[/b] How many times does $01101$ appear as a not necessarily contiguous substring of $0101010101010101$? (Stated another way, how many ways can we choose digits from the second string, such that when read in order, these digits read $01101$?)


[b]p10.[/b] A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself. For example, $28$ is a perfect number because $1 + 2 + 4 + 7 + 14 = 28$. Let $n_i$ denote the ith smallest perfect number. Define $$f(x) =\sum_{i|n_x}\sum_{j|n_i}\frac{1}{j}$$  (where $\sum_{i|n_x}$ means we sum over all positive integers $i$ that are divisors of $n_x$). Compute $f(2)$, given there are at least $50 $perfect numbers.


[b]p11.[/b] Let $O$ be a circle with chord $AB$. The perpendicular bisector to $AB$ is drawn, intersecting $O$ at points $C$ and $D$, and intersecting $AB$ at the midpoint $E$. Finally, a circle $O'$ with diameter $ED$ is drawn, and intersects the chord $AD$ at the point $F$. Given $EC = 12$, and $EF = 7$, compute the radius of $O$.


[b]p12.[/b] Suppose $r$, $s$, $t$ are the roots of the polynomial $x^3 - 2x + 3$. Find $$\frac{1}{r^3 - 2}+\frac{1}{s^3 - 2}+\frac{1}{t^3 - 2}.$$


[b]p13.[/b] Let $a_1$, $a_2$,..., $a_{14}$ be points chosen independently at random from the interval $[0, 1]$. For $k = 1$, $2$,$...$, $7$, let $I_k$ be the closed interval lying between $a_{2k-1}$ and $a_{2k}$ (from the smaller to the larger). What is the probability that the intersection of $I_1$, $I_2$,$...$, $I_7$ is nonempty?


[b]p14.[/b] Consider all triangles $\vartriangle ABC$ with area $144\sqrt3$ such that $$\frac{\sin A \sin B \sin C}{
\sin A + \sin B + \sin C}=\frac14.$$ Over all such triangles $ABC$, what is the smallest possible perimeter?


[b]p15.[/b] Let $N$ be the number of sequences $(x_1,x_2,..., x_{2018})$ of elements of $\{1, 2,..., 2019\}$, not necessarily distinct, such that $x_1 + x_2 + ...+ x_{2018}$ is divisible by $2018$. Find the last three digits of $N$.


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve this problem, we need to use the extended law of sines and some properties of triangles. Let's go through the steps in detail.

1. **Using the Extended Law of Sines:**
   The extended law of sines states that for any triangle \( \triangle ABC \) with circumradius \( R \):
   \[
   a = 2R \sin A, \quad b = 2R \sin B, \quad c = 2R \sin C
   \]
   where \( a, b, c \) are the sides opposite to angles \( A, B, C \) respectively.

2. **Substituting into the Given Expression:**
   We are given:
   \[
   \frac{\sin A \sin B \sin C}{\sin A + \sin B + \sin C} = \frac{1}{4}
   \]
   Substituting \( a = 2R \sin A \), \( b = 2R \sin B \), and \( c = 2R \sin C \) into the expression, we get:
   \[
   \frac{(2R \sin A)(2R \sin B)(2R \sin C)}{8R^3 \left( \frac{a}{2R} + \frac{b}{2R} + \frac{c}{2R} \right)} = \frac{1}{4}
   \]
   Simplifying, we have:
   \[
   \frac{8R^3 \sin A \sin B \sin C}{8R^3 (\sin A + \sin B + \sin C)} = \frac{1}{4}
   \]
   \[
   \frac{\sin A \sin B \sin C}{\sin A + \sin B + \sin C} = \frac{1}{4}
   \]
   This confirms the given condition.

3. **Relating \( abc \) and \( R \):**
   From the above, we can write:
   \[
   \frac{abc}{a + b + c} = R^2
   \]

4. **Using the Area of the Triangle:**
   The area \( [ABC] \) of the triangle can be expressed as:
   \[
   [ABC] = \frac{1}{2} ab \sin C
   \]
   Since \( \sin C = \frac{c}{2R} \), we have:
   \[
   [ABC] = \frac{1}{2} ab \cdot \frac{c}{2R} = \frac{abc}{4R}
   \]
   Also, the area can be expressed using the inradius \( r \) and the semiperimeter \( s \):
   \[
   [ABC] = rs
   \]
   Equating the two expressions for the area, we get:
   \[
   \frac{abc}{4R} = rs
   \]

5. **Relating \( R \) and \( r \):**
   From the above, we have:
   \[
   \frac{abc}{a + b + c} = 2Rr
   \]
   Given \( \frac{abc}{a + b + c} = R^2 \), we get:
   \[
   R^2 = 2Rr \implies R = 2r
   \]

6. **Using Euler's Theorem:**
   Euler's theorem for the distance \( d \) between the circumcenter and the incenter states:
   \[
   d^2 = R^2 - 2Rr = R(R - 2r)
   \]
   Since \( d^2 \geq 0 \), we have:
   \[
   R \geq 2r
   \]
   Therefore, the minimum is achieved when \( R = 2r \), which implies \( R = r \).

7. **Finding the Side Length and Perimeter:**
   When \( R = r \), the triangle is equilateral. The area of an equilateral triangle with side length \( x \) is:
   \[
   \frac{x^2 \sqrt{3}}{4}
   \]
   Given the area is \( 144\sqrt{3} \), we solve for \( x \):
   \[
   \frac{x^2 \sqrt{3}}{4} = 144 \sqrt{3} \implies x^2 = 576 \implies x = 24
   \]
   The perimeter of the equilateral triangle is:
   \[
   3 \times 24 = 72
   \]

The final answer is \( \boxed{72} \)