[b]p1.[/b] Let $a$ and $b$ be complex numbers such that $a^3 + b^3 = -17$ and $a + b = 1$. What is the value of $ab$?


[b]p2.[/b] Let $AEFB$ be a right trapezoid, with $\angle AEF = \angle EAB = 90^o$. The two diagonals $EB$ and $AF$ intersect at point $D$, and $C$ is a point on $AE$ such that $AE \perp DC$. If $AB = 8$ and $EF = 17$, what is the lenght of $CD$?


[b]p3.[/b] How many three-digit numbers $abc$ (where each of $a$, $b$, and $c$ represents a single digit, $a \ne 0$) are there such that the six-digit number $abcabc$ is divisible by $2$, $3$, $5$, $7$, $11$, or $13$?


[b]p4.[/b] Let $S$ be the sum of all numbers of the form $\frac{1}{n}$ where $n$ is a postive integer and $\frac{1}{n}$ terminales in base $b$, a positive integer. If $S$ is $\frac{15}{8}$, what is the smallest such $b$?


[b]p5.[/b] Sysyphus is having an birthday party and he has a square cake that is to be cut into $25$ square pieces. Zeus gets to make the first straight cut and messes up badly. What is the largest number of pieces Zeus can ruin (cut across)? Diagram?


[b]p6.[/b] Given $(9x^2 - y^2)(9x^2 + 6xy + y^2) = 16$ and $3x + y = 2$. Find $x^y$.


[b]p7.[/b] What is the prime factorization of the smallest integer $N$ such that $\frac{N}{2}$ is a perfect square, $\frac{N}{3}$ is a perfect cube, $\frac{N}{5}$ is a perfect fifth power?


[b]p8.[/b] What is the maximum number of pieces that an spherical watermelon can be divided into with four straight planar cuts?


[b]p9.[/b] How many ordered triples of integers $(x,y,z)$ are there such that $0 \le x, y, z \le 100$ and $$(x - y)^2 + (y - z)^2 + (z - x)^2 \ge (x + y - 2z) + (y + z - 2x)^2 + (z + x - 2y)^2.$$


[b]p10.[/b] Find all real solutions to $(2x - 4)^2 + (4x - 2)^3 = (4x + 2x - 6)^3$.


[b]p11.[/b] Let $f$ be a function that takes integers to integers that also has $$f(x)=\begin{cases} x - 5\,\, if \,\, x \ge 50 \\ f (f (x + 12)) \,\, if \,\, x < 50 \end{cases}$$ Evaluate $f (2) + f (39) + f (58).$


[b]p12.[/b] If two real numbers are chosen at random (i.e. uniform distribution) from the interval $[0,1]$, what is the probability that theit difference will be less than $\frac35$?


[b]p13.[/b] Let $a$, $b$, and $c$ be positive integers, not all even, such that $a < b$, $b = c - 2$, and $a^2 + b^2 = c^2$. What is the smallest possible value for $c$?


[b]p14.[/b] Let $ABCD$ be a quadrilateral whose diagonals intersect at $O$. If $BO = 8$, $OD = 8$, $AO = 16$, $OC = 4$, and $AB = 16$, then find $AD$.


[b]p15.[/b] Let $P_0$ be a regular icosahedron with an edge length of $17$ units. For each nonnegative integer $n$, recursively construct $P_{n+1}$ from Pn by performing the following procedure on each face of $P_n$: glue a regular tetrahedron to that face such that three of the vertices of the tetrahedron are the midpoints of the three adjacent edges of the face, and the last vertex extends outside of $P_n$. Express the number of square units in the surface area of $P_{17}$ in the form $$\frac{u^v\cdot w \sqrt{x}}{y^z}$$	, where $u, v, w, x, y$, and $z$ are integers, all greater than or equal to $2$, that satisfy the following conditions: the only perfect square that evenly divides $x$ is $1$, the GCD of $u$ and y is $1$, and neither $u$ nor $y$ divides $w$. Answers written in any other form will not be considered correct!


PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Given the quadrilateral \(ABCD\) with diagonals intersecting at \(O\), we know the following:
   - \(BO = 8\)
   - \(OD = 8\)
   - \(AO = 16\)
   - \(OC = 4\)
   - \(AB = 16\)

2. We start by analyzing \(\triangle ABO\). Since \(AB = AO = 16\) and \(BO = 8\), we can place the points in a coordinate system for simplicity:
   - Let \(A = (0, 0)\)
   - Let \(B = (16, 0)\)
   - Since \(O\) is the midpoint of \(BD\), we need to find the coordinates of \(O\).

3. Using the distance formula, we can find the coordinates of \(O\):
   - Since \(O\) is the midpoint of \(BD\), we have \(O = \left(\frac{16 + x}{2}, \frac{0 + y}{2}\right)\).
   - Given \(BO = 8\), we can write the equation:
     \[
     \sqrt{(x - 16)^2 + y^2} = 8
     \]
   - Given \(AO = 16\), we can write the equation:
     \[
     \sqrt{x^2 + y^2} = 16
     \]

4. Solving these equations simultaneously:
   - From \(\sqrt{x^2 + y^2} = 16\), we get:
     \[
     x^2 + y^2 = 256
     \]
   - From \(\sqrt{(x - 16)^2 + y^2} = 8\), we get:
     \[
     (x - 16)^2 + y^2 = 64
     \]

5. Substituting \(y^2 = 256 - x^2\) into \((x - 16)^2 + y^2 = 64\):
   \[
   (x - 16)^2 + (256 - x^2) = 64
   \]
   \[
   x^2 - 32x + 256 + 256 - x^2 = 64
   \]
   \[
   -32x + 512 = 64
   \]
   \[
   -32x = -448
   \]
   \[
   x = 14
   \]

6. Substituting \(x = 14\) back into \(x^2 + y^2 = 256\):
   \[
   14^2 + y^2 = 256
   \]
   \[
   196 + y^2 = 256
   \]
   \[
   y^2 = 60
   \]
   \[
   y = \sqrt{60} = 2\sqrt{15}
   \]

7. Thus, the coordinates of \(O\) are \((14, 2\sqrt{15})\).

8. Since \(O\) is the midpoint of \(BD\), we can find the coordinates of \(D\):
   - Let \(D = (x_D, y_D)\)
   - Since \(O\) is the midpoint, we have:
     \[
     \left(\frac{16 + x_D}{2}, \frac{0 + y_D}{2}\right) = (14, 2\sqrt{15})
     \]
   - Solving for \(x_D\) and \(y_D\):
     \[
     \frac{16 + x_D}{2} = 14 \implies 16 + x_D = 28 \implies x_D = 12
     \]
     \[
     \frac{0 + y_D}{2} = 2\sqrt{15} \implies y_D = 4\sqrt{15}
     \]

9. Therefore, the coordinates of \(D\) are \((12, 4\sqrt{15})\).

10. Finally, we calculate the distance \(AD\):
    \[
    AD = \sqrt{(12 - 0)^2 + (4\sqrt{15} - 0)^2}
    \]
    \[
    AD = \sqrt{12^2 + (4\sqrt{15})^2}
    \]
    \[
    AD = \sqrt{144 + 240}
    \]
    \[
    AD = \sqrt{384}
    \]
    \[
    AD = 8\sqrt{6}
    \]

The final answer is \(\boxed{8\sqrt{6}}\)