MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2012 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685241p23295291]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. Let

O

and

I

represent the center of the circumscribed and inscribed circle, respectively inside on a triangle with side length

3

4

, and

5

. The length of

OI

is ...
p2. Let

x, y

and

z

be prime numbers that satisfy the equation

34x- 51y = 2012z

. The value of

x + y + z

is ...
p3. It is known that four dice are balanced and different, each of which is in the form of a regular octagon with with numbers

1, 2, 3, ..., 8

at faces. The four dice are tossed (tossed) together once. The probability that there are two dice with the same number of points is ...
p4. The real value functions

f

and

g

each have the equations

f(x)=\sqrt{\lfloor x- \rfloor - a} \,\,\, , \,\,\, g(x)=\sqrt{x^2 - \frac{x\sqrt2}{\sqrt{a}}}

where

a

is positive integer. If the domain

gof

\{x|3 \frac1 2\le x <4\}

, then the numbers of

a

are ...
p5. Given a prime number

p> 2

. If

S

is the sum of all natural numbers n that if

n^2 + pn

is the square of an integer number then

S =

...
p6. For any real number

x

defined

\{x\}

as the integer nearest to

x

. for example

\{1,9\} = 2

\{ 0, 501\} = 1

, etc. If

n

is a positive integer number multiple of 2012, the number of many positive integers

k

that satisfy

\{\sqrt[3]{k}\}

is ...
p7. A large number of natural numbers

n <100

that have multiples of the form

123456789123456789... 123456789

is...
p8. Given a parallelogram

ABCD

. The point

M

AB

is such that

AM/AB = 0, 017

, and point

N

AD

so that

AN/AD = 17/2009

. Let

AC \cap MN = P

, then

AC/AP =

...
p9. In a meeting,

5

married couples will be seated at a round table. How many ways are there to arrange the sitting positions of the

5

married couples in such a way, such that exactly

3

husbands sit next to their wives?
p10. If

p, q

and

r

are the roots of

x^3 - x^2 + x - 2 = 0

, then

p^3 + q^3 + r^3 =

....
p11. If

m

and

n

are positive integers that satisfy

m^2 + n^5 = 252

, then

m + n =

...
p12. At

\vartriangle ABC

, point

D

lies on the line

BC

BC = 3

\angle ABC = 30^o

, and

\angle ADC = 45^o

. Lentgh of

AC

is ...
p13. Five students,

A,B,C,D,E

are in one group in the relay race. If

A

is not can run first and

D

cannot run last, then the number of possible arrangements is ...
p14. It is known that

H

is the set of all natural numbers less than

2012

whose prime factors are not more than

3

. Next defined set

S =\left\{ \frac{1}{n}| n \in H \right\}.

x

is the sum of all the elements of

S

, then

\lfloor x \rfloor =

...
p15. Given two circles

\Gamma_1

and

\Gamma_2

which intersect at two points, namely

A

and

B

with

AB = 10

. The line segment joining the centers of the two circles intersects the circle

\Gamma_1

and

\Gamma_2

at P and Q, respectively. If

PQ = 3

and the radius of the circle

\Gamma_1

13

, then the radius of the circle

\Gamma_2

is ...
p16. The number of pairs of integers

(x, y)

that satisfy

\frac{1}{x}+ \frac{1}{y}- \frac{1}{xy^3}= \frac{3}{4}

is ...
p17. For positive real numbers

x

and

y

with

xy = \frac13

, the minimum value of

\frac{1}{9x^6} + \frac{1}{4y^6}

is ...
p18. The number of pairs of positive integers

(a, b)

that satisfy

4^a + 4a^2 + 4 = b^2

is ..
p19. Given a triangle

ABC

, the length of

AB

is equal to twice the length of

AC

. Suppose

D

and

E

in segments

AB

and

BC

, respectively, so

\angle BAE = \angle ACD

. If

F = AE\cap CD

and

CEF

is an equilateral triangle, then the measure of the angles of triangle

ABC

is ......
p20. The number of positive integers

n

that satisfy

n\le 2012

and is a square number perfect or cubic or to the power of

4

or to the power of

5

or ... or to the power of

10

, is...

Problem Statement