MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2008 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684687p23289130]here
Time: 90 minutes

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Write only the answers to the questions given.

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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.

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Each question is worth 1 (one) point.
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

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}

Problem Statement