MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2013 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685416p23297006]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. Given three circles with radius

r = 2

, which are tangent to each other. The total area of the three circles and their bounded area is equal to...
p2.

2013

lights are controlled by

2013

switch buttons which are numbered

1

2

3

...

2013

. Pressing the switch button once will change the light on (on or off). At first all the lights were off. On the first day, all the switch buttons are pressed once. On the second day, all switch buttons numbered

2

or multiples of

2

are pressed once. By doing the same thing on day

n

, all light switch buttons numbered

n

or multiples of

n

are pressed once. And so on. How many lights were on after the operation on day

2013

was performed?
p3. Given a real function

f

with

f(x)=\frac{cx}{2x-3}

x\ne \frac32

and

f(f(x))=x

for all

x\ne \frac32

. The value of

f(2013)

is ...
p4. Pairs of positive integers

(x, y)

that satisfy

\frac{xy^2}{x+y}

prime number are ...
p5. If

|x|+x+y=10

and

x+|y|-y = 12

, then the value of

x + y

is ...
p6. The number of positive integers

n

, that satisfy

n^2 - 660

is a perfect square number, is...
p7. How many sequences of nine terms are

a_1

a_2

...

a_9

, each of which is

0

1

2

3

4

5

6

7

8

, or

9

, and contains exactly one sequence

a_i

a_j

where

a_i

is even and

a_j

is odd?
p8. Natural number

n

is said to be "beautiful" if

n

consists of

3

different digits or more and the constituent digits form an arithmetic sequence or geometric sequence. For example, number

123

is beautiful because

1, 2, 3

form an arithmetic sequence. There are ... beautiful numbers .
p9. Let

M

be the midpoint of side

BC

in triangle

ABC

and

\angle CAB = 45^o

\angle ABC = 30^o

, then

\tan\angle AMC

is ...
p10. Given a prime number

p > 2013

. Suppose

a

and

b

are natural numbers so that

a + b

is divisible by

p

but not divisible by

p^2

. If it is known that

a^{2013} + b^{2013}

is divisible by

p^2

then the number of natural numbers

n\le 2013

so that

a^{2013} + b^{2013}

is divisible by

p^n

is ...
p11. There are six kindergarten children each bringing a meal. They will hold a cross gift, where the food is collected and then divided again so that each child receives food that is not the food that was brought back. The number of ways to do this is...
p12. The graphs of the parabola

y = x^2 - a

and

x = y^2 - b

with

a > 0

and

b > 0

, intersect at four points

(x_1, y_1)

(x_2, y_2)

(x_3, y_3)

, and

(x_4, y_4)

. The value of

(x_1 + x_2)(x_1 + x_3)(x_1 + x_4)

is ...
p13. A dice is tossed

2

(two) times. Let

a

and

b

be the numbers that appear on the first and second tosses, respectively. The probability that there are real numbers

x

y

, and

z

that satisfy the equations

x + y + z = a

and

x^2 + y^2 + z^2 = b

is ...
p14. Let

\Delta_1

\Delta_2

\Delta_3

, be a sequence of equilateral triangles whose side length

\Delta_1

1

. For

n\ge 1

, triangle

\Delta_{n+1}

is defined in the following way: first define

P_n

as a square whose vertices the angle lies on the sides of

\Delta_n

, then defined

L_n

as the largest circle in

P_n

, then defined

\Delta_{n+1}

, an equilateral triangle whose vertices lie on the circumference of the circle. The side length of

\Delta_{2013}

is ...
p15. A sequence

x_1.x_2, ...,x_n, ...

is defined by:

x_1=2

and

x_{n+1}=\left(1+\frac{1}{n}\right) x_n + \frac{2}{n}

for every natural number

n

. The value of

x_{2013}

is ...
p16. Given a square with side lengths equal to

2\sqrt3

. Inside the square there are two equilateral triangles whose bases are opposite sides of the square. The intersection of the two equilateral triangles forms a rhombus. The area of the rhombus is equal to...
p17. The number of positive integers

a

and

b

that satisfy gcd

(a, b) = 1

and

\frac{a}{b}+\frac{25b}{21a}

is an integer , is ...
p18. Given triangle

ABC

AB = 20

AC = 21

and

BC = 29

. The points

D

and

E

lie on the segment

BC

, so

BD = 8

and

EC = 9

. The measure of

\angle DAE

is equal to ...
p19. A competition is attended by

20

participants. In each round, two participants compete. Each participant who loses twice is removed from the competition, the last participant in the competition is the winner. If it is known that the winner of the competition has never lost, the number of matches held in that competition is...
p20. The sum of all the integers

x

, such that

^2 \log (x^2 - 4x - 1)

is an integer, is ...

Problem Statement