
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.** There are  $32$  balls in a box:  $6$  are blue,  $8$  are red,  $4$  are yellow, and  $14$  are brown. If I pull out three balls at once, what is the probability that none of them are brown?**p2.** Circles  $A$  and  $B$  are concentric, and the area of circle  $A$  is exactly  $20\%$  of the area of circle  $B$ . The circumference of circle  $B$  is  $10$ . A square is inscribed in circle  $A$ . What is the area of that square?**p3.** If  $x^2 +y^2 = 1$  and  $x, y \in R$ , let  $q$  be the largest possible value of  $x+y$  and  $p$  be the smallest possible value of  $x + y$ . Compute  $pq$ .**p4.** Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches  $11$  points before the other person can reach  $10$  points, then the person who reached  $11$  points wins. If instead the score ends up being tied  $10$ -to- $10$ , then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is  $70\%$  and the score is currently at  $9$ -to- $9$ . What is the probability that Yizheng wins the game?**p5.** The squares on an  $8\times 8$  chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is  $\#1$ , the top-right square is  $\#8$ , and the bottom-right square is  $\#64$ ).  $1$  grain of wheat is placed on square  $\#1$ ,  $2$  grains on square  $\#2$ ,  $4$  grains on square  $\#3$ , and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column?**p6.** Let  $f$  be any function that has the following property: For all real numbers  $x$  other than  $0$  and  $1$ ,   $$ f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2. $$   Compute  $f(2)$ .**p7.** Find all solutions of:    $$ (x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x. $$  **p8.** Let  $\vartriangle ABC$  be a triangle where  $AB = 25$  and  $AC = 29$ .  $C_1$  is a circle that has  $AB$  as a diameter and  $C_2$  is a circle that has  $BC$  as a diameter.  $D$  is a point on  $C_1$  so that  $BD = 15$  and  $CD = 21$ .  $C_1$  and  $C_2$  clearly intersect at  $B$ ; let  $E$  be the other point where  $C_1$  and  $C_2$  intersect. Find all possible values of  $ED$ .


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

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