
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.** What is the maximum possible value of  $m$  such that there exist  $m$  integers  $a_1, a_2, ..., a_m$  where all the decimal representations of  $a_1!, a_2!, ..., a_m!$  end with the same amount of zeros?**p2.** Let  $f : R \to R$  be a function such that  $f(x) + f(y^2) = f(x^2 + y)$ , for all  $x, y \in R$ . Find the sum of all possible  $f(-2017)$ .**p3.** What is the sum of prime factors of  $1000027$ ?**p4.** Let   $$ \frac{1}{2!} +\frac{2}{3!} + ... +\frac{2016}{2017!} =\frac{n}{m}, $$   where  $n, m$  are relatively prime. Find  $(m - n)$ .**p5.** Determine the number of ordered pairs of real numbers  $(x, y)$  such that  $\sqrt[3]{3 - x^3 - y^3} =\sqrt{2 - x^2 - y^2}$ **p6.** Triangle  $\vartriangle ABC$  has  $\angle B = 120^o$ ,  $AB = 1$ . Find the largest real number  $x$  such that  $CA - CB > x$  for all possible triangles  $\vartriangle ABC$ .**p7.**Jung and Remy are playing a game with an unfair coin. The coin has a probability of  $p$  where its outcome is heads. Each round, Jung and Remy take turns to flip the coin, starting with Jung in round  $ 1$ . Whoever gets heads first wins the game. Given that Jung has the probability of  $8/15$  , what is the value of  $p$ ?**p8.** Consider a circle with  $7$  equally spaced points marked on it. Each point is  $ 1$  unit distance away from its neighbors and labelled  $0,1,2,...,6$  in that order counterclockwise. Feng is to jump around the circle, starting at the point  $0$  and making six jumps counterclockwise with distinct lengths  $a_1, a_2, ..., a_6$  in a way such that he will land on all other six nonzero points afterwards. Let  $s$  denote the maximum value of  $a_i$ . What is the minimum possible value of  $s$ ?**p9.**Justin has a  $4 \times 4 \times 4$  colorless cube that is made of  $64$  unit-cubes. He then colors  $m$  unit-cubes such that none of them belong to the same column or row of the original cube. What is the largest possible value of  $m$ ?**p10.** Yikai wants to know Liang’s secret code which is a  $6$ -digit integer  $x$ . Furthermore, let  $d(n)$  denote the digital sum of a positive integer  $n$ . For instance,  $d(14) = 5$  and  $d(3) = 3$ . It is given that   $$ x + d(x) + d(d(x)) + d(d(d(x))) = 999868. $$   Please find  $x$ .


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