
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.** What is the remainder when  $2021$  is divided by  $102$ ?**p2.** Brian has  $2$  left shoes and  $2$  right shoes. Given that he randomly picks  $2$  of the  $4$  shoes, the probability he will get a left shoe and a right shoe is  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers. Compute  $p + q$ .**p3.** In how many ways can  $59$  be written as a sum of two perfect squares? (The order of the two perfect squares does not matter.)


<u>Round 2</u>**p4.** Two positive integers have a sum of  $60$ . Their least common multiple is  $273$ . What is the positive diffeerence between the two numbers?**p5.** How many ways are there to distribute  $13$  identical apples among  $4$  identical boxes so that no two boxes receive the same number of apples? A box may receive zero apples.**p6.** In square  $ABCD$  with side length  $5$ ,  $P$  lies on segment  $AB$  so that  $AP = 3$  and  $Q$  lies on segment  $AD$  so that  $AQ = 4$ . Given that the area of triangle  $CPQ$  is  $x$ , compute  $2x$ .


<u>Round 3</u>**p7.** Find the number of ordered triples  $(a, b, c)$  of nonnegative integers such that  $2a+3b+5c = 15$ .**p8.** What is the greatest integer  $n \le 15$  such that  $n + 1$  and  $n^2 + 3$  are both prime?**p9.** For positive integers  $a, b$ , and  $c$ , suppose that  $gcd \,\,(a, b) = 21$ ,  $gcd \,\,(a, c) = 10$ , and  $gcd \,\,(b,c) = 11$ . Find  $\frac{abc}{lcm \,\,(a,b,c)}$  . (Note:  $gcd$  is the greatest common divisor function and  $lcm$  is the least common multiple function.)


<u>Round 4</u>**p10.** The vertices of a square in the coordinate plane are at  $(0, 0)$ ,  $(0, 6)$ ,  $(6, 0)$ , and  $(6, 6)$ . Line  $\ell$  intersects the square at exactly two lattice points (that is, points with integer coordinates). How many such lines  $\ell$  are there that divide the square into two regions, one of them having an area of  $12$ ?**p11.** Let  $f(n)$  be defined as follows for positive integers  $n$ :  $f(1) = 0$ ,  $f(n) = 1$  if  $n$  is prime, and  $f(n) = f(n - 1) + 1$  otherwise. What is the maximum value of  $f(n)$  for  $n \le 120$ ?**p12.** The graph of the equation  $y = x^3 + ax^2 + bx + c$  passes through the points  $(2,4)$ ,  $(3, 9)$ , and  $(4, 16)$ . What is  $b$ ?


PS. You should use hide for answers. Rounds 5- 8 have been posted [here](https://artofproblemsolving.com/community/c3h2949415p26408227). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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