
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>Set 1</u>**B1.** Evaluate  $2 + 0 - 2 \times 0$ .**B2.** It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses?**B3.** Let  $a$  be the answer to this question. What is  $\frac{1}{2-a}$ ?


<u>Set 2</u>**B4.** Every day at Andover is either sunny or rainy. If today is sunny, there is a  $60\%$  chance that tomorrow is sunny and a  $40\%$  chance that tomorrow is rainy. On the other hand, if today is rainy, there is a  $60\%$  chance that tomorrow is rainy and a  $40\%$  chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is  $n$ ?**B5.** In the diagram below, what is the value of  $\angle DD'Y$  in degrees?
![Image](https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png)**B6.** Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? 

Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid.


<u>Set 3</u>**B7.** Let  $P$  be a point on side  $AB$  of square  $ABCD$  with side length  $8$  such that  $PA = 3$ . Let  $Q$  be a point on side  $AD$  such that  $P Q \perp P C$ . The area of quadrilateral  $PQDB$  can be expressed in the form  $m/n$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m + n$ .**B8.** Jessica and Jeffrey each pick a number uniformly at random from the set  $\{1, 2, 3, 4, 5\}$  (they could pick the same number). If Jessica’s number is  $x$  and Jeffrey’s number is  $y$ , the probability that  $x^y$  has a units digit of  $1$  can be expressed as  $m/n$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .**B9.** For two points  $(x_1, y_1)$  and  $(x_2, y_2)$  in the plane, we define the taxicab distance between them as  $|x_1 - x_2| + |y_1 - y_2|$ . For example, the taxicab distance between  $(-1, 2)$  and  $(3,\sqrt2)$  is  $6-\sqrt2$ . What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same?


<u>Set 4</u>**B10.** Will wants to insert some × symbols between the following numbers:   $$ 1\,\,\,2\,\,\,3\,\,\,4\,\,\,6 $$   to see what kinds of answers he can get. For example, here is one way he can insert  $\times$  symbols:   $$ 1 \times 23 \times 4 \times 6 = 552. $$   Will discovers that he can obtain the number  $276$ . What is the sum of the numbers that he multiplied together to get  $276$ ?**B11.** Let  $ABCD$  be a parallelogram with  $AB = 5$ ,  $BC = 3$ , and  $\angle BAD = 60^o$  . Let the angle bisector of  $\angle ADC$  meet  $AC$  at  $E$  and  $AB$  at  $F$ . The length  $EF$  can be expressed as  $m/n$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m + n$ ?**B12.** Find the sum of all positive integers  $n$  such that  $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$ .

Note:  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .


<u>Set 5</u>**B13.** This year, February  $29$  fell on a Saturday. What is the next year in which February  $29$  will be a Saturday?**B14.** Let  $f(x) = \frac{1}{x} - 1$ . Evaluate   $$ f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) . $$  **B15.** Square  $WXYZ$  is inscribed in square  $ABCD$  with side length  $1$  such that  $W$  is on  $AB$ ,  $X$  is on  $BC$ ,  $Y$  is on  $CD$ , and  $Z$  is on  $DA$ . Line  $W Y$  hits  $AD$  and  $BC$  at points  $P$  and  $R$  respectively, and line  $XZ$  hits  $AB$  and  $CD$  at points  $Q$  and  $S$  respectively. If the area of  $WXYZ$  is  $\frac{13}{18}$  , then the area of  $PQRS$  can be expressed as  $m/n$  for relatively prime positive integers  $m$  and  $n$ . What is  $m + n$ ?



PS. You had better use hide for answers. Last sets have been posted [here](https://artofproblemsolving.com/community/c4h2777424p24371574). Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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