
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
<details><summary>B stands for Bernoulli, G stands for Germain</summary>they had two problem sets under those two names</details> **B1** What is the sum of the first  $5$  positive integers?**B2** Bread picks a number  $n$ . He finds out that if he multiplies  $n$  by  $23$  and then subtracts  $20$ , he gets  $46279$ . What is  $n$ ?**B3** A *Harshad* Number is a number that is divisible by the sum of its digits. For example,  $27$  is divisible by  $2 + 7 = 9$ . Only one two-digit multiple of  $9$  is not a *Harshad* Number. What is this number?**B4 / G1** There are  $5$  red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen?**B5** Let  $a$  be a  $1$ -digit positive integer and  $b$  be a  $3$ -digit positive integer. If the product of  $a$  and  $b$  is a $ 4$ -digit integer, what is the minimum possible value of the sum of  $a$  and  $b$ ?**B6 / G2** A circle has radius  $6$ . A smaller circle with the same center has radius  $5$ . What is the probability that a dart randomly placed inside the outer circle is outside the inner circle?**B7** Call a two-digit integer “sus” if its digits sum to  $10$ . How many two-digit primes are sus?**B8  / G3** Alex and Jeff are playing against Max and Alan in a game of tractor with  $2$  standard decks of  $52$  cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the  $5$ s are worth  $5$  points, the  $10$ s are worth  $10$  points, and the kings are worth 10 points. Given that a team needs  $50$  percent more points than the other to win, what is the minimal score Alan and Max need to win?**B9  / G4**  Bob has a sandwich in the shape of a rectangular prism. It has side lengths  $10$ ,  $5$ , and  $5$ . He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece?**B10  / G5** Aven makes a rectangular fence of area  $96$  with side lengths  $x$  and  $y$ . John makesva larger rectangular fence of area 186 with side lengths  $x + 3$  and  $y + 3$ . What is the value of  $x + y$ ?**B11  / G6** A number is prime if it is only divisible by itself and  $1$ . What is the largest prime number  $n$  smaller than  $1000$  such that  $n + 2$  and  $n - 2$  are also prime? 
Note:  $1$  is not prime.**B12  / G7** Sally has  $3$  red socks,  $1$  green sock,  $2$  blue socks, and  $4$  purple socks. What is the probability she will choose a pair of matching socks when only choosing  $2$  socks without replacement?**B13  / G8** A triangle with vertices at  $(0, 0)$ , $ (3, 0)$ ,  $(0, 6)$  is filled with as many  $1 \times 1$  lattice squares as possible. How much of the triangle’s area is not filled in by the squares?**B14  / G10** A series of concentric circles  $w_1, w_2, w_3, ...$  satisfy that the radius of  $w_1 = 1$  and the radius of  $w_n =\frac34$  times the radius of  $w_{n-1}$ . The regions enclosed in  $w_{2n-1}$  but not in  $w_{2n}$  are shaded for all integers  $n > 0$ . What is the total area of the shaded regions?**B15  / G12**  $10$  cards labeled 1 through  $10$  lie on a table. Kevin randomly takes  $3$  cards and Patrick randomly takes 2 of the remaining  $7$  cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card?**G9** Let  $A$  and  $B$  be digits. If  $125A^2 + B161^2 = 11566946$ . What is  $A + B$ ?**G11** How many ordered pairs of integers  $(x, y)$  satisfy  $y^2 - xy + x = 0$ ?**G13**  $N$  consecutive integers add to  $27$ . How many possible values are there for  $N$ ?**G14** A circle with center O and radius  $7$  is tangent to a pair of parallel lines  $\ell_1$  and  $\ell_2$ . Let a third line tangent to circle  $O$  intersect  $\ell_1$  and  $\ell_2$  at points  $A$  and  $B$ . If  $AB = 18$ , find  $OA + OB$ .**G15** Let   $$  M =\prod ^{42}_{i=0}(i^2 - 5). $$   Given that  $43$  doesn’t divide  $M$ , what is the remainder when M is divided by  $43$ ?


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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