MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2009 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684692p23289182]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.

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to be more exact:

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in years 2002-08 time was 90' for part A and 120' for part B

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since years 2009 time is 210' for part A and B totally

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each problem in part A is 1 point, in part B is 7 points
p1. Three black, red, and white dice are rolled together. Write the results of the throw so that the sum of the three dice is

8

.
p2. The number of real numbers x that satisfy the equation

x^4- 2x^3 + 5x^2-176x +-2009 = 0

is ...
p3. The rational numbers

a < b < c

form a arithmetic sequence and

\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3

. The number of positive numbers a that satisfies is ...
p4. Let

N

represent the set of all positive integers and

S= \{n\in N| \frac{n^{2009}+2}{n+1} \in N\}

. The number of subsets of

S

is ...
p5. Given a triangle

ABC

with

\tan \angle CAB = \frac{22}{7}

. Through the vertex

A

, a line is drawn such that it divides the side

BC

into segments of lengths

3

and

17

. The area of triangle

ABC

is ....
p6. The minimum value of

f(x)=\frac{9x^2\sin^2x+4}{x\sin x}

for

0 < x <\pi

is ...
p7. Given a triangle whose lengths from the three altitudes of the triangle are integers. If the lengths of the two lines of altitudes are

10

and

6

, then the maximum length of the third line is ...
p8. A function

f : Z \to Q

has the property

f(x+1)=\frac{1+f(x)}{1-f(x)}

for every

x\in Z

. If

f(2) = 2

, then the value of the function

f(2009)

is ...
p9. It is known that a right triangle

ABC

has side lengths

a, b

, and

c

and

a < b < c

. Let

r

and

R

represent the lengths of the inradii and the circumradii, respectively. If

\frac{r(a+b+b)}{R^2}=\sqrt3

then the value of

\frac{r}{a+b+b}

is ...
p10. If

\tan x + \tan y = 25

and

\cot x + \cot y = 30

, then the value of

\tan (x + y)

is
p11. On the right of

100!

there are digits equal to

0

in a row as many as ...
p12. There are four pairs of shoes, four shoes will be drawn at random. The probability that a pair is drawn is ...
p13. It is known that

k, m

, and

n

are three positive integers that satisfy

\frac{k}{m}+\frac{m}{4n}=\frac{1}{6}

. The smallest number

m

that satisfies is ...
p14. The number of prime numbers

p

that satisfy

(2p-1)^3 + (3p)^2 = 6^p

is ...
p15. If

x_1, x_2, ..., x_{2009}

are real numbers, then the smallest value of

\cos x1 \sin x_2 + \cos x_2 \sin x_3 + ...+ \cos x_{2009} \sin x_1

is ...
p16. Let

a, b, c

be the roots of the polynomial

x^3-8x2 + 4x-2

. If

f(x) = x^3 + px^2 + qx + r

is a polynomial with roots

a + b - c

b + c - a

c + a - b

then

f(1) = ....

p17. The number of obtuse triangles with a natural number side that has the longest side

10

is ... Note: two congruent triangles are considered equal.
p18. Let

n

be the smallest natural number that has exactly

2009

factors and

n

is a multiple of

2009

. The smallest prime factor of

n

is ...
p19. Let

p(x) = x^2-6

and

A = \{x\in R | p(p(x)) = x\}

. The maximum value of

\{|x| : x\in A\}

is ...
p20. Let

q=\frac{\sqrt5+1}{2}

and

[x]

represent the largest integer that is less than or equal to

x

. The value of

[q[qn]] - [q^2n]

for any

n\in N

is ...

Problem Statement