
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.** Prair takes some set  $S$  of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from  $1$  to  $10$  is on her list! What is the smallest possible value of  $|S|$ , the number of elements in her set  $S$ ?**p2.** Jake has  $2021$  balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs?**p3.** Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to  $(2, 2)$ , but she doesn’t know how to get there. So each second, she rides one unit in the positive  $x$  or  $y$ -direction, each with probability  $\frac12$  . If the probability that she makes it to  $(2, 2)$  during her ride can be expressed as  $\frac{a}{b}$  for positive integers  $a, b$  with  $gcd(a, b) = 1$ , then find  $a + b$ .**p4.** Triangle  $ABC$  with  $AB = BC = 6$  and  $\angle ABC = 120^o$  is rotated about  $A$ , and suppose that the images of points  $B$  and  $C$  under this rotation are  $B'$  and  $C'$ , respectively. Suppose that  $A$ ,  $B'$  and  $C$  are collinear in that order. If the area of triangle  $B'CC'$  can be expressed as  $a - b\sqrt{c}$  for positive integers  $a, b, c$  with csquarefree, find  $a + b + c$ .**p5.** Find the sum of all possible values of  $a + b + c + d$  if  $a, b, c, $ d are positive integers satisfying  $$ ab + cd = 100, $$    $$ ac + bd = 500. $$  **p6.** Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points  $(1, 2)$  and  $(2, 0)$  and a chute-ladder between points  $(1, 3)$  and  $(4, 0)$ , whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from  $(0, 0)$  to  $(4, 4)$ ?**p7.** There are  $8$  identical cubes that each belong to  $8$  different people. Each person randomly picks a cube. The probability that exactly  $3$  people picked their own cube can be written as  $\frac{a}{b}$  , where  $a$  and  $b$  are positive integers with  $gcd(a, b) = 1$ . Find  $a + b$ .**p8.** Suppose that  $p(R) = Rx^2 + 4x$  for all  $R$ . There exist finitely many integer values of  $R$  such that  $p(R)$  intersects the graph of  $x^3 + 2021x^2 + 2x + 1$  at some point  $(j, k)$  for integers  $j$  and  $k$ . Find the sum of all possible values of  $R$ .**p9.** Let  $a, b, c$  be the roots of the polynomial  $x^3 - 20x^2 + 22$ . Find  $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$ .**p10.** In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank  $n \times n$  grid with probability  $k$ ; he notices that the expected value of the score of the resulting grid is equal to  $k$ , too! Given that  $k > 0.9999$ , find the minimum possible value of  $n$ .**p11.** Find the sum of all  $x$  from  $2$  to  $1000$  inclusive such that   $$ \prod^x_{n=2} \log_{n^n}(n + 1)^{n+2} $$   is an integer.**p12.** Let triangle  $ABC$  with incenter  $I$  and circumcircle  $\Gamma$  satisfy  $AB = 6\sqrt3$ ,  $BC = 14$ ,  and  $CA = 22$ . Construct points  $P$  and  $Q$  on rays  $BA$  and  $CA$  such that  $BP = CQ = 14$ . Lines  $PI$  and  $QI$  meet the tangents from  $B$  and  $C$  to  $\Gamma$ , respectively, at points  $X$  and  $Y$  . If  $XY$  can be expressed as  $a\sqrt{b}-c$  for positive integers  $a, b, c$  with  $c$  squarefree, find  $a + b + c$ .


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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