
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
<u>Round 1</u>**p1.** Today, the date  $4/9/16$  has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property?**p2.** What is the greatest integer less than  $100$  whose digit sumis equal to its greatest prime factor?**p3.** In chess, a bishop can only move diagonally any number of squares. Find the number of possible squares a bishop starting in a corner of a  $20\times 16$  chessboard can visit in finitely many moves, including the square it stars on.

<u>Round 2</u>**p4.** What is the fifth smallest positive integer with at least  $5$  distinct prime divisors?**p5.** Let  $\tau (n)$  be the number of divisors of a positive integer  $n$ , including  $1$  and  $n$ . Howmany positive integers  $n \le 1000$  are there such that  $\tau (n) > 2$  and  $\tau (\tau (n)) = 2$ ?**p6.** How many distinct quadratic polynomials  $P(x)$  with leading coefficient  $1$  exist whose roots are positive integers and whose coefficients sum to  $2016$ ?


<u>Round 3</u>**p7.** Find the largest prime factor of  $112221$ .**p8.** Find all ordered pairs of positive integers  $(a,b)$  such that  $\frac{a^2b^2+1}{ab-1}$  is an integer.**p9.** Suppose  $f : Z \to Z$  is a function such that  $f (2x)= f (1-x)+ f (1-x)$  for all integers  $x$ . Find the value of  $f (2) f (0) +f (1) f (6)$ .


<u>Round  4</u>**p10.** For any six points in the plane, what is the maximum number of isosceles triangles that have three of the points as vertices?**p11.** Find the sum of all positive integers  $n$  such that  $\sqrt{n+ \sqrt{n -25}}$  is also a positive integer.**p12.** Distinct positive real numbers are written at the vertices of a regular  $2016$ -gon. On each diagonal and edge of the  $2016$ -gon, the sum of the numbers at its endpoints is written. Find the minimum number of distinct numbers that are now written, including the ones at the vertices.


PS. You should use hide for answers. Rounds 5-8 have been posted   [here](https://artofproblemsolving.com/community/c3h3158474p28715078). and 9-12 [here](https://artofproblemsolving.com/community/c3h3162282p28763571). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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