Indonesia Regional [hide=MO]also know as provincial level, is a qualifying round for National Math Olympiad[/hide]  
Year 2006  [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684669p23288980]here[/url][/hide]

Time: 90 minutes [hide=Rules]
$\bullet$ Write only the answers to the questions given. 
$\bullet$ Some questions can have more than one correct answer. You are asked to provide the most correct or exact answer to a question like this. Scores will only be given to the giver of the most correct or most exact answer.
$\bullet$ Each question is worth 1 (one) point.[/hide]


p1. The sum of all integers between $\sqrt[3]{2006}$ and  $\sqrt{2006}$ is ...

p2. In trapezoid $ABCD$, side $AB$ is parallel to $DC$. A circle tangent to all four sides of the trapezoid can be drawn. If $AB = 75$ and $DC = 40$, then the perimeter of the trapezoid $ABCD = ...$

p3. The set of all $x$ that satisfies $(x-1)^3 + (x-2)^2 = 1$ is ...

p4. The largest two-digit prime number which is the sum of two other prime numbers is ...

p5. Afkar selects the terms of an infinite geometric sequence $1, \frac12, \frac14, \frac18$, to create a new infinite geometric sequence whose sum is $\frac17$. The first three terms Afkar chooses are ...

p6. The area of ​​the sides of a cuboid are $486$, $486$, $243$, $243$, $162$, $162$. The volume of the cuboid is ...

p7. The maximum value of the function $f(x) = \left(\frac13 \right)^{x^2-4x+3}$ is ...

p8. Given the function $f(x) = ||x - 2| - a| -3$. If the graph f intersects the $x$-axis at exactly three points, then $a =. ..$

p9. For natural numbers $n$, write $s(n) = 1 + 2 + ...+ n$ and $p(n) = 1 \times 2 \times ... \times n$. The smallest even number $n$ such that $p(n)$ is divisible by $s(n)$ is ...

p10. If $|x|+ x + y = 10$ and $x + |y|-y = 12$, then $x + y = ...$

p11. A set of three natural numbers is called an [i]arithmetic set [/i] if one of its elements is the average of the other two elements. The number of arithmetic subsets of $\{1,2,3,...,8\}$ is ...

p12. From each one-digit number $a$, the number $N$ is made by juxtaposing the three numbers $a + 2$, $a + 1$, $a$ i.e. $N = \overline{(a+2)(a+1)a}$ . For example, for $a = 8$, $N = 1098$. The ten such numbers $N$ have the greatest common divisor  ...

p13. If $x^2+\frac{1}{x^2}=47$, then $\sqrt{x}+\frac{1}{\sqrt{x}}= ...$ 

p14. A class will choose a student from among them to represent the class. Every student has the same opportunity to be selected. The probability that a male student is selected is $\frac23$ times the probability that a female student is selected. The percentage of male students in the class is ...

p15. In triangle $ABC$, the bisector of angle $A$ intersects side $BC$ at point D. If $AB = AD = 2$ and $BD = 1$, then $CD = ...$

p16. If $(x-1)^2$ divides $ax^4 + bx^3 + 1$, then $ab = ...$

p17. From point $O$, two half-lines (rays) $l_1$ and $l_2$ are drawn which form an acute angle . The different points $A_1, A_3, A_5$ lie on the line $l_2$, while the points $A_2, A_4, A_6$ lie on the line $l_1$. If $A_1A_2 = A_2A_3 = A_3A_4 = A_4O = OA_5 = A_5A_6 = A_6A_1$, then = ...

p18. The number of different $7$-digit numbers that can be formed by changing the order of the numbers $2504224$ is ...

p19. Evan creates a sequence of natural numbers $a_1, a_2, a_3$, which satisfies $a_{k+1}-a_k = 2(a_k-a_{k-1})-1$, for $k = 2, 3,...,$ and $a_2 - a_1 = 2$. If $2006$ appears in the sequence, the smallest possible value of $a_1$ is ...

p20. In triangle $ABC$, the medians from vertex $B$ and vertex $C$ intersect at right angles to each other. The minimum value of $\cot B + \cot C$ is ...
1. We need to find the sum of all integers between $\sqrt[3]{2006}$ and $\sqrt{2006}$.
   - First, calculate $\sqrt[3]{2006} \approx 12.6$ and $\sqrt{2006} \approx 44.8$.
   - The integers between these values are $13, 14, \ldots, 44$.
   - The sum of an arithmetic series is given by $S = \frac{n}{2} (a + l)$, where $n$ is the number of terms, $a$ is the first term, and $l$ is the last term.
   - Here, $a = 13$, $l = 44$, and $n = 44 - 13 + 1 = 32$.
   - Therefore, the sum is $S = \frac{32}{2} (13 + 44) = 16 \times 57 = 912$.

The final answer is $\boxed{912}$.