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

Time: 90 minutes <details><summary>Rules</summary>$\bullet$  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.</details>


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 *arithmetic set* 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 ...

Present the answer in LaTex format: \boxed{Your answer}
user