
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
*20 problems for 25 minutes.***p1.** Evaluate  $20 \times 21 + 2021$ .**p2.** Let points  $A$ ,  $B$ ,  $C$ , and  $D$  lie on a line in that order. Given that  $AB = 5CD$  and  $BD = 2BC$ , compute  $\frac{AC}{BD}$ .**p3.** There are  $18$  students in Vincent the Bug's math class. Given that  $11$  of the students take U.S. History,  $15$  of the students take English, and  $2$  of the students take neither, how many students take both U.S. History and English?**p4.** Among all pairs of positive integers  $(x, y)$  such that  $xy = 12$ , what is the least possible value of  $x + y$ ?**p5.** What is the smallest positive integer  $n$  such that  $n! + 1$  is composite?**p6.** How many ordered triples of positive integers  $(a, b,c)$  are there such that  $a + b + c = 6$ ?**p7.** Thomas orders some pizzas and splits each into  $8$  slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the  $29$  other math club students. What is the fewest number of pizzas that Thomas could have ordered?**p8.** Stephanie has two distinct prime numbers  $a$  and  $b$  such that  $a^2-9b^2$  is also a prime. Compute  $a + b$ .**p9.** Let  $ABCD$  be a unit square and  $E$  be a point on diagonal  $AC$  such that  $AE = 1$ . Compute  $\angle BED$ , in degrees.**p10.** Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube.**p11.** In base  $b$ ,  $130_b$  is equal to  $3n$  in base ten, and  $1300_b$  is equal to  $n^2$  in base ten. What is the value of  $n$ , expressed in base ten?**p12.** Lin is writing a book with  $n$  pages, numbered  $1,2,..., n$ . Given that  $n > 20$ , what is the least value of  $n$  such that the average number of digits of the page numbers is an integer?**p13.** Max is playing bingo on a  $5\times 5$  board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win?**p14.** Given that  $x$  and  $y$  are distinct real numbers such that  $x^2 + y = y^2 + x = 211$ , compute the value of  $|x - y|$ .**p15.** How many ways are there to place 8 indistinguishable pieces on a  $4\times 4$  checkerboard such that there are two pieces in each row and two pieces in each column?**p16.** The Manhattan distance between two points  $(a, b)$  and  $(c, d)$  in the plane is defined to be  $|a - c| + |b - d|$ . Suppose Neil, Neel, and Nail are at the points  $(5, 3)$ ,  $(-2,-2)$  and  $(6, 0)$ , respectively, and wish to meet at a point  $(x, y)$  such that their Manhattan distances to $ (x, y)$  are equal. Find  $10x + y$ .**p17.** How many positive integers that have a composite number of divisors are there between  $1$  and  $100$ , inclusive?**p18.** Find the number of distinct roots of the polynomial   $$ (x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90) $$  .**p19.** In triangle  $ABC$ , let  $D$  be the foot of the altitude from  $ A$  to  $BC$ . Let  $P,Q$  be points on  $AB$ ,  $AC$ , respectively, such that  $PQ$  is parallel to  $BC$  and  $\angle PDQ = 90^o$ . Given that  $AD = 25$ ,  $BD = 9$ , and  $CD = 16$ , compute  $111 \times PQ$ .**p20.** The simplified fraction with numerator less than  $1000$  that is closest but not equal to  $\frac{47}{18}$  is  $\frac{p}{q}$  , where  $p$  and  $q$  are relatively prime positive integers. Compute  $p$ .


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

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