
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.** Trung took five tests this semester. For his first three tests, his average was  $60$ , and for the fourth test he earned a  $50$ . What must he have earned on his fifth test if his final average for all five tests was exactly  $60$ ?**p2.** Find the number of pairs of integers  $(a, b)$  such that  $20a + 16b = 2016 - ab$ .**p3.** Let  $f : N \to N$  be a strictly increasing function with  $f(1) = 2016$  and  $f(2t) = f(t) + t$  for all  $t \in N$ . Find  $f(2016)$ .**p4.** Circles of radius  $7$ ,  $7$ ,   $18$ , and  $r$  are mutually externally tangent, where  $r = \frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Find  $m + n$ .**p5.** A point is chosen at random from within the circumcircle of a triangle with angles  $45^o$ ,  $75^o$ ,  $60^o$ . What is the probability that the point is closer to the vertex with an angle of  $45^o$  than either of the two other vertices?**p6.** Find the largest positive integer  $a$  less than  $100$  such that for some positive integer  $b$ ,  $a - b$  is a prime number and  $ab$  is a perfect square.**p7.** There is a set of  $6$  parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds  $6$  more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made.**p8.** Triangle  $ABC$  has sides  $AB = 5$ ,  $AC = 4$ , and  $BC = 3$ . Let  $O$  be any arbitrary point inside  $ABC$ , and  $D \in BC$ ,  $E \in AC$ ,  $F \in AB$ , such that  $OD \perp BC$ ,  $OE \perp AC$ ,  $OF \perp AB$ . Find the minimum value of  $OD^2 + OE^2 + OF^2$ .**p9.** Find the root with the largest real part to  $x^4-3x^3+3x+1 = 0$  over the complex numbers.**p10.** Tony has a board with  $2$  rows and  $4$  columns. Tony will use  $8$  numbers from  $1$  to  $8$  to fill in this board, each number in exactly one entry. Let array  $(a_1,..., a_4)$  be the first row of the board and array  $(b_1,..., b_4)$  be the second row of the board. Let  $F =\sum^{4}_{i=1}|a_i - b_i|$ , calculate the average value of  $F$  across all possible ways to fill in.


PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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