[b]p1.[/b] A right triangle has a hypotenuse of length $25$ and a leg of length $16$. Compute the length of the other leg of this triangle.


[b]p2.[/b] Tanya has a circular necklace with $5$ evenly-spaced beads, each colored red or blue. Find the number of distinct necklaces in which no two red beads are adjacent. If a necklace can be transformed into another necklace through a series of rotations and reflections, then the two necklaces are considered to be the same.


[b]p3.[/b] Find the sum of the digits in the decimal representation of $10^{2016} - 2016$.


[b]p4.[/b] Let $x$ be a real number satisfying $$x^1 \cdot x^2 \cdot x^3 \cdot x^4 \cdot x^5 \cdot x^6 = 8^7.$$ Compute the value of $x^7$.


[b]p5.[/b] What is the smallest possible perimeter of an acute, scalene triangle with integer side lengths?


[b]p6.[/b] Call a sequence $a_1, a_2, a_3,..., a_n$ mountainous if there exists an index $t$ between $1$ and $n$ inclusive such that $$a_1 \le a_2\le ... \le a_t \,\,\,\, and \,\,\,\, a_t \ge a_{t+1} \ge ... \ge a_n$$
In how many ways can Bishal arrange the ten numbers $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$, $5$, and $5$ into a mountainous sequence? (Two possible mountainous sequences are $1$, $1$, $2$, $3$, $4$, $4$, $5$, $5$, $3$, $2$ and $5$, $5$, $4$, $4$, $3$, $3$, $2$, $2$, $1$, $1$.)


[b]p7.[/b] Find the sum of the areas of all (non self-intersecting) quadrilaterals whose vertices are the four points $(-3,-6)$, $(7,-1)$, $(-2, 9)$, and $(0, 0)$.



[b]p8.[/b] Mohammed Zhang's favorite function is $f(x) =\sqrt{x^2 - 4x + 5} +\sqrt{x^2 + 4x + 8}$. Find the minumum possible value of $f(x)$ over all real numbers $x$.


[b]p9.[/b] A segment $AB$ with length $1$ lies on a plane. Find the area of the set of points $P$ in the plane for which $\angle APB$ is the second smallest angle in triangle $ABP$.


[b]p10.[/b] A binary string is a dipalindrome if it can be produced by writing two non-empty palindromic strings one after the other. For example, $10100100$ is a dipalindrome because both $101$ and $00100$ are palindromes. How many binary strings of length $18$ are both palindromes and dipalindromes?


PS. You should use hide for answers.  Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the minimum value of the function \( f(x) = \sqrt{x^2 - 4x + 5} + \sqrt{x^2 + 4x + 8} \), we can interpret the terms geometrically.

1. Rewrite the terms inside the square roots:
   \[
   \sqrt{x^2 - 4x + 5} = \sqrt{(x-2)^2 + 1^2}
   \]
   \[
   \sqrt{x^2 + 4x + 8} = \sqrt{(x+2)^2 + 2^2}
   \]
   This allows us to interpret the terms as distances in the coordinate plane.

2. Define points based on these distances:
   \[
   A = (x, 0), \quad B = (2, 1), \quad C = (-2, 2)
   \]
   The expression \( f(x) \) represents the sum of the distances from point \( A \) to points \( B \) and \( C \).

3. Reflect point \( B \) over the \( x \)-axis to get \( B' \):
   \[
   B' = (2, -1)
   \]
   The distance \( AB' \) is the same as \( AB \) because the reflection does not change the distance.

4. The sum of the distances \( AB' + AC \) will be minimized when point \( A \) is collinear with points \( B' \) and \( C \).

5. Find the equation of the line passing through \( B' \) and \( C \):
   \[
   \text{slope of } B'C = \frac{2 - (-1)}{-2 - 2} = \frac{3}{-4} = -\frac{3}{4}
   \]
   The equation of the line in point-slope form is:
   \[
   y - (-1) = -\frac{3}{4}(x - 2)
   \]
   Simplifying, we get:
   \[
   y + 1 = -\frac{3}{4}x + \frac{3}{2}
   \]
   \[
   y = -\frac{3}{4}x + \frac{1}{2}
   \]

6. Find the \( x \)-intercept of this line (where \( y = 0 \)):
   \[
   0 = -\frac{3}{4}x + \frac{1}{2}
   \]
   \[
   \frac{3}{4}x = \frac{1}{2}
   \]
   \[
   x = \frac{1}{2} \cdot \frac{4}{3} = \frac{2}{3}
   \]

7. Evaluate \( f(x) \) at \( x = \frac{2}{3} \):
   \[
   f\left(\frac{2}{3}\right) = \sqrt{\left(\frac{2}{3} - 2\right)^2 + 1^2} + \sqrt{\left(\frac{2}{3} + 2\right)^2 + 2^2}
   \]
   \[
   = \sqrt{\left(-\frac{4}{3}\right)^2 + 1} + \sqrt{\left(\frac{8}{3}\right)^2 + 4}
   \]
   \[
   = \sqrt{\frac{16}{9} + 1} + \sqrt{\frac{64}{9} + 4}
   \]
   \[
   = \sqrt{\frac{16}{9} + \frac{9}{9}} + \sqrt{\frac{64}{9} + \frac{36}{9}}
   \]
   \[
   = \sqrt{\frac{25}{9}} + \sqrt{\frac{100}{9}}
   \]
   \[
   = \frac{5}{3} + \frac{10}{3}
   \]
   \[
   = \frac{15}{3} = 5
   \]

Thus, the minimum value of \( f(x) \) is \( \boxed{5} \).