Indonesia Regional [hide=MO]also know as provincial level, is a qualifying round for National Math Olympiad[/hide]  
Year 2005  [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c4h2671390p23150609]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. If $a$ is a rational number and $b$ is an irrational number, then $a + b$ is number ...

p2. The sum of the first ten prime numbers is ...

p3. The number of sets $X$ that satisfy $\{1, 2\} \subseteq X \subseteq \{1, 2, 3, 4, 5\}$ is ...

p4. If $N = 123456789101112...9899100$, then the first three numbers of $\sqrt{N}$ are ...

p5. Suppose $ABCD$ is a trapezoid with $BC\parallel AD$. The points $ P$ and $R$ are the midpoints of sides $AB$ and $CD$, respectively. The point $Q$ is on the side $BC$ so that $BQ : QC = 3 : 1$, while the point $S$ is on the side $AD$ so that $AS : SD = 1: 3$. Then the ratio of the area of ​​the quadrilateral $PQRS$ to the area of ​​the trapezoid $ABCD$ is ...

p6. The smallest three-digit number that is a perfect square and a perfect cube (to the power of three) at the same time is ...

p7. If $a, b$ are two natural numbers $a\le b$ so that $\frac{\sqrt3+\sqrt{a}}{\sqrt3+\sqrt{b}}$ is a rational number, then the ordered pair $(a, b) = ...$

p8. If $AB = AC$, $AD = BD$, and the angle $\angle DAC = 39^o$, then the angle $\angle BAD$ is ...
[img]https://cdn.artofproblemsolving.com/attachments/1/1/8f0bcd793f0de025081967ce4259ea75fabfcb.png[/img]

p9. When climbing a hill, a person walks at a speed of $1\frac12$ km/hour. As he descended the hill, he walked three times as fast. If the time it takes to travel back and forth from the foot of the hill to the top of the hill and back to the foot of the hill is $6$ hours, then the distance between the foot of the hill and the top of the hill (in km) is ...

p10. A regular hexagon and an equilateral triangle have the same perimeter. If the area of ​​the triangle is $\sqrt3$, then the area of ​​the hexagon is ...

p11. Two dice are thrown simultaneously. The probability that the sum of the two numbers that appear is prime is ..

p12. The perimeter of an equilateral triangle is $p$. Let $Q$ be a point in the triangle. If the sum of the distances from $Q$ to the three sides of the triangle is $s$, then, expressed in terms of $s$, $p = ...$

p13. The sequence of natural numbers $(a, b, c)$ with $a\ge b \ge c$, which satisfies both equations $ab + bc = 44$ and $ac + bc = 23$ is ...

p14. Four distinct points lie on a line. The distance between any two points can be sorted into the sequence $1, 4, 5, k, 9, 10$. Then $k =$ ...

p15. A group consists of $2005$ members. Each member holds exactly one secret. Each member can send a letter to any other member to convey all the secrets he holds. The number of letters that need to be sent for all the group members to know the whole secret is ...

p16. The number of pairs of integers $(x, y)$ that satisfy the equation $2xy-5x + y = 55$ is ...

p17. The sets A and B are independent and $A\cup B = \{1, 2, 3,..., 9\}$. The product of all elements of $A$ is equal to the sum of all elements of $B$. The smallest element of $B$ is ...

p18. The simple form of $\frac{(2^3-1)(3^3-1)(4^3-1)...(100^3-1)}{(2^3+1)(3^3+1)(4^3+1)...(100^3+1)}$ is ...

p19. Suppose $ABCD$ is a regular triangular pyramid, which is a four-sided space that is in the form of an equilateral triangle. Let $S$ be the midpoint of edge $AB$ and $T$ the midpoint of edge $CD$. If the length of the side $ABCD$ is $1$ unit length, then the length of $ST$ is ...

p20. For any real number $a$, the notation$[a]$ denotes the largest integer less than or equal to $a$. If $x$ is a real number that satisfies $[x+\sqrt3]=[x]+[\sqrt3]$, then $x -[x]$  will not be greater than ...
1. If \( a \) is a rational number and \( b \) is an irrational number, then \( a + b \) is irrational. This is because the sum of a rational number and an irrational number is always irrational. 

   To see why, assume for contradiction that \( a + b \) is rational. Then we can write:
   \[
   a + b = r \quad \text{(where \( r \) is rational)}
   \]
   Rearranging gives:
   \[
   b = r - a
   \]
   Since \( r \) and \( a \) are both rational, \( r - a \) is rational. This contradicts the assumption that \( b \) is irrational. Therefore, \( a + b \) must be irrational.

2. The sum of the first ten prime numbers is:
   \[
   2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 = 129
   \]

3. The number of sets \( X \) that satisfy \( \{1, 2\} \subseteq X \subseteq \{1, 2, 3, 4, 5\} \) is calculated as follows:
   - \( X \) must include \( \{1, 2\} \).
   - The remaining elements to choose from are \( \{3, 4, 5\} \).
   - Each of these elements can either be included or not included in \( X \).

   Therefore, the number of possible sets \( X \) is:
   \[
   2^3 = 8
   \]

The final answer is \( \boxed{8} \)