[u]Round 5[/u]


[b]5.1[/b] A circle with a radius of $1$ is inscribed in a regular hexagon. This hexagon is inscribed in a larger circle. If the area that is outside the hexagon but inside the larger circle can be expressed as $\frac{a\pi}{b} - c\sqrt{d}$, where $a, b, c, d$ are positive integers, $a, b$ are relatively prime, and no prime perfect square divides into $d$. find the value of $a + b + c + d$.

[b]5.2[/b] At a dinner party, $10$ people are to be seated at a round table. If person A cannot be seated next to person $B$ and person $C$ must be next to person $D$, how many ways can the 10 people be seated? Consider rotations of a configuration identical.

[b]5.3[/b] Let $N$ be the sum of all the positive integers that are less than $2022$ and relatively prime to $1011$. Find $\frac{N}{2022}$.

[u]Round 6[/u]


[b]6.1[/b] The line $y = m(x - 6)$ passes through the point $ A$ $(6, 0)$, and the line $y = 8 -\frac{x}{m}$ pass through point $B$ $(0,8)$. The two lines intersect at point $C$. What is the largest possible area of triangle $ABC$?

[b]6.2[/b] Let $N$ be the number of ways there are to arrange the letters of the word MATHEMATICAL such that no two As can be adjacent. Find the last $3$ digits of $\frac{N}{100}$.

[b]6.3[/b] Find the number of ordered triples of integers $(a, b, c)$ such that $|a|, |b|, |c| \le 100$ and $3abc = a^3 + b^3 + c^3$.


[u]Round 7[/u]


[b]7.1[/b] In a given plane, let $A, B$ be points such that $AB = 6$. Let $S$ be the set of points such that for any point $C$ in $S$, the circumradius of $\vartriangle ABC$ is at most $6$. Find $a + b + c$ if the area of $S$ can be expressed as $a\pi + b\sqrt{c}$ where $a, b, c$ are positive integers, and $c$ is not divisible by the square of any prime.

[b]7.2[/b] Compute $\sum_{1\le a<b<c\le 7} abc$.

[b]7.3[/b] Three identical circles are centered at points $A, B$, and $C$ respectively and are drawn inside a unit circle. The circles are internally tangent to the unit circle and externally tangent to each other. A circle centered at point $D$ is externally tangent to circles $A, B$, and $C$. If a circle centered at point $E$ is externally tangent to circles $A, B$, and $D$, what is the radius of circle $E$? The radius of circle $E$ can be expressed as $\frac{a\sqrt{b}-c}{d}$ where $a, b, c$, and d are all positive integers, gcd(a, c, d) = 1, and b is not divisible by the square of any prime. What is the sum of $a + b + c + d$?


[u]Round 8[/u]


[b]8.[/b] Let $A$ be the number of unused Algebra problems in our problem bank. Let $B$ be the number of times the letter ’b’ appears in our problem bank. Let M be the median speed round score. Finally, let $C$ be the number of correct answers to Speed Round $1$. 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/c3h2826128p24988676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to find the sum of all positive integers less than \(2022\) that are relatively prime to \(1011\). We will use the Principle of Inclusion/Exclusion (PIE) to achieve this.

1. **Prime Factorization of \(1011\)**:
   \[
   1011 = 3 \times 337
   \]

2. **Sum of all positive integers less than \(2022\)**:
   The sum of the first \(2021\) positive integers is given by:
   \[
   \sum_{k=1}^{2021} k = \frac{2021 \times 2022}{2} = 1011 \times 2022
   \]

3. **Sum of integers divisible by \(3\)**:
   The number of integers less than \(2022\) that are divisible by \(3\) is:
   \[
   \left\lfloor \frac{2021}{3} \right\rfloor = 673
   \]
   The sum of these integers is:
   \[
   3 \times \sum_{k=1}^{673} k = 3 \times \frac{673 \times 674}{2} = 3 \times 226051 = 678153
   \]

4. **Sum of integers divisible by \(337\)**:
   The number of integers less than \(2022\) that are divisible by \(337\) is:
   \[
   \left\lfloor \frac{2021}{337} \right\rfloor = 5
   \]
   The sum of these integers is:
   \[
   337 \times \sum_{k=1}^{5} k = 337 \times \frac{5 \times 6}{2} = 337 \times 15 = 5055
   \]

5. **Sum of integers divisible by \(1011\)**:
   The number of integers less than \(2022\) that are divisible by \(1011\) is:
   \[
   \left\lfloor \frac{2021}{1011} \right\rfloor = 1
   \]
   The sum of these integers is:
   \[
   1011 \times 1 = 1011
   \]

6. **Applying the Principle of Inclusion/Exclusion**:
   The sum of integers less than \(2022\) that are not relatively prime to \(1011\) is:
   \[
   678153 + 5055 - 1011 = 682197
   \]

7. **Sum of integers relatively prime to \(1011\)**:
   The sum of all integers less than \(2022\) is \(1011 \times 2022\). Subtracting the sum of integers not relatively prime to \(1011\):
   \[
   1011 \times 2022 - 682197 = 2044042 - 682197 = 1361845
   \]

8. **Finding \(\frac{N}{2022}\)**:
   \[
   \frac{1361845}{2022} = 672
   \]

The final answer is \(\boxed{672}\).