[u]Round 5[/u]

[b]5.1.[/b] Quadrilateral $ABCD$ is such that $\angle ABC = \angle ADC = 90^o$ , $\angle BAD = 150^o$ , $AD = 3$, and $AB = \sqrt3$. The area of $ABCD$ can be expressed as $p\sqrt{q}$ for positive integers $p, q$ where $q$ is not divisible by the square of any prime. Find $p + q$.

[b]5.2.[/b] Neetin wants to gamble, so his friend Akshay describes a game to him. The game will consist of three dice: a $100$-sided one with the numbers $1$ to $100$, a tetrahedral one with the numbers $1$ to $4$, and a normal $6$-sided die. If Neetin rolls numbers with a product that is divisible by $21$, he wins. Otherwise, he pays Akshay $100$ dollars. The number of dollars that Akshay must pay Neetin for a win in order to make this game fair is $a/b$ for relatively prime positive integers $a, b$. Find $a + b$. (Fair means the expected net gain is $0$. )

[b]5.3.[/b] What is the sum of the fourth powers of the roots of the polynomial $P(x) = x^2 + 2x + 3$?


[u]Round 6[/u]

[b]6.1.[/b] Consider the set $S = \{1, 2, 3, 4,..., 25\}$. How many ordered $n$-tuples $S_1 = (a_1, a_2, a_3,..., a_n)$ of pairwise distinct ai exist such that $a_i \in S$ and $i^2 | a_i$ for all $1 \le i \le n$?

[b]6.2.[/b] How many ways are there to place $2$ identical rooks and $ 1$ queen on a $ 4 \times 4$ chessboard such that no piece attacks another piece? (A queen can move diagonally, vertically or horizontally and a rook can move vertically or horizontally)

[b]6.3.[/b] Let $L$ be an ordered list $\ell_1$, $\ell_2$, $...$, $\ell_{36}$ of consecutive positive integers who all have the sum of their digits not divisible by $11$. It is given that $\ell_1$ is the least element of $L$. Find the least possible value of $\ell_1$.


[u]Round 7[/u]

[b]7.1.[/b] Spencer, Candice, and Heather love to play cards, but they especially love the highest cards in the deck - the face cards (jacks, queens, and kings). They also each have a unique favorite suit: Spencer’s favorite suit is spades, Candice’s favorite suit is clubs, and Heather’s favorite suit is hearts. A dealer pulls out the $9$ face cards from every suit except the diamonds and wants to deal them out to the $3$ friends. How many ways can he do this so that none of the $3$ friends will see a single card that is part of their favorite suit?

[b]7.2.[/b] Suppose a sequence of integers satisfies the recurrence $a_{n+3} = 7a_{n+2} - 14a_{n+1} + 8a_n$. If $a_0 = 4$, $a_1 = 9$, and $a_2 = 25$, find $a_{16}$. Your answer will be in the form $2^a + 2^b + c$, where $2^a < a_{16} < 2^{a+1}$ and $b$ is as large as possible. Find $a + b + c$.

[b]7.3.[/b] Parallel lines $\ell_1$ and $\ell_2$ are $1$ unit apart. Unit square $WXYZ$ lies in the same plane with vertex $W$ on $\ell_1$. Line $\ell_2$ intersects segments $YX$ and $YZ$ at points $U$ and $O$, respectively. Given $UO =\frac{9}{10}$, the inradius of $\vartriangle YOU$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$. Find $m + n$.

[u]Round 8[/u]

[b]8.[/b] Let $A$ be the number of contestants who participated in at least one of the three rounds of the 2020 ABMC April contest. Let $B$ be the number of times the letter b appears in the Accuracy Round. Let $M$ be the number of
people who submitted both the speed and accuracy rounds before 2:00 PM EST. Further, let $C$ be the number of
times the letter c appears in the Speed Round. Estimate
$$A \cdot B + M \cdot C.$$Your answer will be scored according to the following formula, where $X$ is the correct answer and $I$ is your input.
$$max \left\{ 0, \left\lceil min \left\{13 - \frac{|I-X|}{0.05 |I|}, 13 - \frac{|I-X|}{0.05 |I-2X|} \right\} \right\rceil \right\}$$


PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2766239p24226402]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To find the sum of the fourth powers of the roots of the polynomial \( P(x) = x^2 + 2x + 3 \), we will use the properties of the roots and symmetric functions.

1. **Identify the roots and their properties:**
   Let the roots of the polynomial \( P(x) = x^2 + 2x + 3 \) be \( r_1 \) and \( r_2 \). By Vieta's formulas, we know:
   \[
   r_1 + r_2 = -2 \quad \text{and} \quad r_1 r_2 = 3
   \]

2. **Express the fourth powers in terms of the roots:**
   We need to find \( r_1^4 + r_2^4 \). We start by expressing \( r_1^4 \) and \( r_2^4 \) in terms of \( r_1 \) and \( r_2 \):
   \[
   r_1^4 + r_2^4 = (r_1^2)^2 + (r_2^2)^2
   \]

3. **Find \( r_1^2 \) and \( r_2^2 \):**
   Using the polynomial equation \( r_1^2 + 2r_1 + 3 = 0 \) and \( r_2^2 + 2r_2 + 3 = 0 \), we solve for \( r_1^2 \) and \( r_2^2 \):
   \[
   r_1^2 = -2r_1 - 3 \quad \text{and} \quad r_2^2 = -2r_2 - 3
   \]

4. **Square the expressions for \( r_1^2 \) and \( r_2^2 \):**
   \[
   r_1^4 = (-2r_1 - 3)^2 = 4r_1^2 + 12r_1 + 9
   \]
   \[
   r_2^4 = (-2r_2 - 3)^2 = 4r_2^2 + 12r_2 + 9
   \]

5. **Sum the squared expressions:**
   \[
   r_1^4 + r_2^4 = (4r_1^2 + 12r_1 + 9) + (4r_2^2 + 12r_2 + 9)
   \]

6. **Substitute \( r_1^2 \) and \( r_2^2 \) back into the equation:**
   \[
   r_1^4 + r_2^4 = 4(-2r_1 - 3) + 12r_1 + 9 + 4(-2r_2 - 3) + 12r_2 + 9
   \]
   Simplify the expression:
   \[
   r_1^4 + r_2^4 = 4(-2r_1 - 3) + 12r_1 + 9 + 4(-2r_2 - 3) + 12r_2 + 9
   \]
   \[
   = (-8r_1 - 12 + 12r_1 + 9) + (-8r_2 - 12 + 12r_2 + 9)
   \]
   \[
   = (-8r_1 + 12r_1 - 12 + 9) + (-8r_2 + 12r_2 - 12 + 9)
   \]
   \[
   = (4r_1 - 3) + (4r_2 - 3)
   \]
   \[
   = 4(r_1 + r_2) - 6
   \]

7. **Use Vieta's formulas to substitute \( r_1 + r_2 \):**
   \[
   r_1 + r_2 = -2
   \]
   \[
   r_1^4 + r_2^4 = 4(-2) - 6
   \]
   \[
   = -8 - 6
   \]
   \[
   = -14
   \]

The final answer is \(\boxed{-14}\).