
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.** How many primes  $p < 100$  satisfy  $p = a^2 + b^2$  for some positive integers  $a$  and  $b$ ?**p2.** For  $a < b < c$ , there exists exactly one Pythagorean triple such that  $a + b + c = 2000$ . Find  $a + c - b$ .**p3.** Five points lie on the surface of a sphere of radius  $ 1$  such that the distance between any two points is at least  $\sqrt2$ . Find the maximum volume enclosed by these five points.**p4.**  $ABCDEF$  is a convex hexagon with  $AB = BC = CD = DE = EF = FA = 5$  and  $AC = CE = EA = 6$ . Find the area of  $ABCDEF$ .**p5.** Joe and Wanda are playing a game of chance. Each player rolls a fair  $11$ -sided die, whose sides are labeled with numbers  $1, 2, ... , 11$ . Let the result of the Joe’s roll be  $X$ , and the result of Wanda’s roll be  $Y$  . Joe wins if  $XY$  has remainder  $ 1$  when divided by  $11$ , and Wanda wins otherwise. What is the probability that Joe wins?**p6.** Vivek picks a number and then plays a game. At each step of the game, he takes the current number and replaces it with a new number according to the following rule: if the current number  $n$  is divisible by  $3$ , he replaces  $n$  with  $\frac{n}{3} + 2$ , and otherwise he replaces  $n$  with  $\lfloor 3 \log_3 n \rfloor$ . If he starts with the number  $3^{2011}$ , what number will he have after  $2011$  steps?

Note that  $\lfloor x\rfloor$  denotes the largest integer less than or equal to  $x$ .**p7.** Define a sequence an of positive real numbers with a $_1 = 1$ , and   $$ a_{n+1} =\frac{4a^2_n - 1}{-2 + \frac{4a^2_n -1}{-2+ \frac{4a^2_n -1}{-2+...}}}. $$  What is  $a_{2011}$ ?**p8.** A set  $S$  of positive integers is called good if for any  $x, y \in S$  either  $x = y$  or  $|x - y| \ge 3$ . How many subsets of  $\{1, 2, 3, ..., 13\}$  are good? Include the empty set in your count.**p9.** Find all pairs of positive integers  $(a, b)$  with  $a \le b$  such that  $10 \cdot lcm \, (a, b) = a^2 + b^2$ . Note that  $lcm \,(m, n)$  denotes the least common multiple of  $m$  and  $n$ .**p10.** For a natural number  $n$ ,  $g(n)$  denotes the largest odd divisor of  $n$ . Find   $$ g(1) + g(2) + g(3) + ... + g(2^{2011}) $$  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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