Indonesia Regional [hide=MO]also know as provincial level, is a qualifying round for National Math Olympiad[/hide]  
Year 2008  [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684687p23289130]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 number of positive divisors from $2008$ is ...

p2. How many ways are there to arrange letters of word MATEMATIKA with the two T's not adjacent? ...

p3. If $0 < b < a$ and $a^2 + b^2 = 6ab$, then $\frac{a+b}{a-b}= ...$

p4. Two of the altitudes of triangle $ABC$ are acute, equal to $4$ and $ 12$, respectively. If the length of the third altitude of the triangle is an integer, then the maximum length of the height of the triangle is ...

p5. In the $XOY$ plane, the number of lines that intersect the $X$ axis at the point with the abscissa of prime numbers and intersect the $Y$ axis at the point with positive integer ordinates and through the point $(4, 3)$ is ...

p6. Given a triangle $ABC$, $AD$ is perpendicular to $BC$ such that $DC = 2$ and $BD = 3$. If $\angle BAC = 45^o$, then the area of ​​triangle $ABC$ is ...

p7. If $x$ and $y$ are integers that satisfy $y^2 + 3x^2y^2 = 30x^2 + 517$, then $3x^2y^2 = ...$

p8. Given a triangle $ABC$, where $BC = a$, $AC = b$ and $\angle C = 60^o$. If $\frac{a}{b}=2+\sqrt3$, then the measure of angle $B$ is ...

p9. One hundred students from a province in Java took part in the selection at the provincial level and the average score was $100$. The number of grade II students who took part in the selection was $50\%$ more than grade III students, and the average score of grade III students was $50\%$ higher of the average score of class II students. The average score of class III students is ...

p10. Given triangle $ABC$, where $BC = 5$, $AC = 12$, and $AB = 13$. Points $D$ and $E$ on $AB$ and $AC$ respectivrly are such that $DE$ divides triangle $ABC$ into two equal parts. The minimum length of $DE$ is ...

p11. Let $a, b, c$ and $d$ be rational numbers. If it is known that the equation $x^4 + ax^3 + bx^2 + cx + d = 0$ has $4$ real roots, two of which are $\sqrt2$ and $\sqrt{2008}$. The value of $a + b + c + d$ is ...

p12. Given a triangle $ABC$ with sides $a, b$, and $c$. The value of $a^2 + b^2 + c^2$ is equal to $16$ times the area of ​​triangle $ABC$. The value of $\cot A + \cot B + \cot C$ is ...

p13. Given $f(x) = x^2 + 4$. Let $x$ and $y$ be positive real numbers that satisfy $f(xy) + f(y-x) = f(y+x)$. The minimum value of $x + y$ is ...

p14. The number of positive integers $n$ less than $2008$ that has exactly $\frac{n}{2}$ numbers less than $n$ and is prime relative to $n$ is ...

p15. A polynomial $f(x)$ satisfies the equation $f(x^2)-x^3f(x) = 2(x^3-1)$ for every $x$ real number. The degree (highest power of  $x$) of $f(x)$ is ..

p16. Assume one year $365$ days. The probability that out of $20$ people chosen at random, two people have a birthday on the same day is ...

p17. Three numbers are chosen at random from $\{1,2,3,...,2008\}$. The probability that the sum of all three is even is ...

p18. Let $|X|$ represent the number of members of the set $X$. If $|A\cup B| = 10$ and $|A| = 4$, then the possible values ​​for $ B$ are ...

p19. It is known that $AD$ is the altitude of triangle $ABC$, $\angle DAB = \angle ACD$, $AD = 6$, $BD = 8$. The area of ​​triangle $ABC$ is ...

p20. Find the value of $\sum_{k=0}^{1004} 3^k {{1004} \choose k}$.
1. Given the equation \(a^2 + b^2 = 6ab\), we start by manipulating this equation to find expressions for \(a+b\) and \(a-b\).

2. First, we rewrite the given equation:
   \[
   a^2 + b^2 = 6ab
   \]

3. We can rearrange this equation to:
   \[
   a^2 + b^2 - 6ab = 0
   \]

4. Recognize that this is a quadratic in terms of \(a\) and \(b\). To solve for \(\frac{a+b}{a-b}\), we need to find expressions for \(a+b\) and \(a-b\).

5. Consider the identity for the square of a binomial:
   \[
   (a-b)^2 = a^2 - 2ab + b^2
   \]
   Using the given equation \(a^2 + b^2 = 6ab\), we substitute \(a^2 + b^2\) in the identity:
   \[
   (a-b)^2 = (a^2 + b^2) - 2ab = 6ab - 2ab = 4ab
   \]
   Therefore:
   \[
   (a-b)^2 = 4ab \implies a-b = 2\sqrt{ab}
   \]

6. Similarly, consider the identity for the square of a sum:
   \[
   (a+b)^2 = a^2 + 2ab + b^2
   \]
   Using the given equation \(a^2 + b^2 = 6ab\), we substitute \(a^2 + b^2\) in the identity:
   \[
   (a+b)^2 = (a^2 + b^2) + 2ab = 6ab + 2ab = 8ab
   \]
   Therefore:
   \[
   (a+b)^2 = 8ab \implies a+b = 2\sqrt{2ab}
   \]

7. Now, we have:
   \[
   a+b = 2\sqrt{2ab} \quad \text{and} \quad a-b = 2\sqrt{ab}
   \]

8. To find \(\frac{a+b}{a-b}\), we divide the expressions:
   \[
   \frac{a+b}{a-b} = \frac{2\sqrt{2ab}}{2\sqrt{ab}} = \frac{\sqrt{2ab}}{\sqrt{ab}} = \sqrt{2}
   \]

Conclusion:
\[
\boxed{\sqrt{2}}
\]