[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] 

[b]B1[/b] What is the sum of the first $5$ positive integers?


[b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$?


[b]B3[/b] A [i]Harshad [/i] 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 [i]Harshad [/i] Number. What is this number?


[b]B4 / G1[/b] 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?


[b]B5[/b] 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 $b$?


[b]B6 / G2[/b] 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?


[b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus?


[b]B8  / G3[/b] 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?


[b]B9  / G4[/b]  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?


[b]B10  / G5[/b] 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 $x + y$?


[b]B11  / G6[/b] 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.


[b]B12  / G7[/b] 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?


[b]B13  / G8[/b] 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?


[b]B14  / G10[/b] 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?


[b]B15  / G12[/b] $10$ 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?


[b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$?


[b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$?


[b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$?


[b]G14[/b] 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 $OA + OB$.


[b]G15[/b] 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 [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. **B1** What is the sum of the first $5$ positive integers?

To find the sum of the first $5$ positive integers, we use the formula for the sum of the first $n$ positive integers:
\[
S = \frac{n(n+1)}{2}
\]
For \( n = 5 \):
\[
S = \frac{5(5+1)}{2} = \frac{5 \cdot 6}{2} = 15
\]

The final answer is \(\boxed{27}\)