
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**p1.** 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.**p2.** 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.**p3.** Find the sum of the digits in the decimal representation of  $10^{2016} - 2016$ .**p4.** 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  $x^7$ .**p5.** What is the smallest possible perimeter of an acute, scalene triangle with integer side lengths?**p6.** 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  $5$ ,  $5$ ,  $4$ ,  $4$ ,  $3$ ,  $3$ ,  $2$ ,  $2$ ,  $1$ ,  $1$ .)**p7.** 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  $(0, 0)$ .**p8.** 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  $x$ .**p9.** 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  $ABP$ .**p10.** 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 [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

Present the answer in LaTex format: \boxed{Your answer}
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