MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2011 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685237p23295278]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 an isosceles triangle

ABC

with

AB = AC

. Let the bisector of the angle

ABC

intersect

AC

at point

D

BC = BD+AD

. The measure of angle

\angle CAB

is ...
p2. If

n

is natural and

\frac12 + \frac13 + \frac15+ \frac{1}{n}

is an integer, then the positive divisor of

n

is as much as ...
p3. If

a\ge b > 1

, then the largest possible value for

^a \log_{\left( \frac{a}{b}\right)}+^b \log_{\left( \frac{b}{a}\right)}

is ...
p4. Given the quadrilateral

ABCD

. All points

A, B, C

and

D

will be numbered

1, 2, 3, 4, 5

6

so that every two points that lie on one side of the number

4

are different. The number of ways of numbering in this way there are as many as ...
p5. Given a function

f

with

f(x) = \sqrt{ax^2 + x}

. All possible values of

a

such that the domain and the range of

f

are the same, are ...
p6. The number of different natural numbers

a, b, c

and

d

that are less than

10

and satisfies the equation

a + b = c + d

are as many as ...
p7. If the two roots of the equation

x^2-2013x + k = 0

are prime numbers, then the possible value of the

k

is ...
p8. If

\left(1- \tan^2\frac{x}{2^{2011}}\right) \left(1- \tan^2\frac{x}{2^{2010}}\right) ... \left(1- \tan^2\frac{x}{2}\right)=2^{2011}\sqrt3 \tan\frac{x}{2^{2011}}

then

\sin 2x

is ...
p9. In Cartesian space, we want to move from point

(2, 0, 11)

to point

(20, 1, 1)

always at coordinates

(x, y, z)

where at least two of

x, y

and

z

are integer numbers and the shortest path. How many ways are there to do this? ...
p10. Let

x, y

and

z

be positive real numbers with the property

xyz = 1

. The smallest value of

(x + 2y)(y + 2z)(xz + 1)

is reached when when

x + y + z

is ...
p11. In the figure below, the lengths of

AE = x

EC = y

and

DC = 2BD

. The ratio of lengths of

BF

and

FE

expressed in terms of

x

and

y

is ... https://cdn.artofproblemsolving.com/attachments/f/3/29a3755bcd481159f144c0058c08a0b6c52a11.png
p12. How may three-digit numbers are there such that all digits are different and the last digit is the sum of the other two digits ? ...
p13. Given a sequence of rational numbers

\{a_k\}_{k\in N}

defined by

a_1 = 2

and

a_{n+1} = \frac{a_n-1}{a_n + 1},\,\,\, n \in N.

The value of

a_{2011}

is ...
p14. Let

\Gamma

be the circumcircle of triangle

ABC

. The chord

AD

is bisects

\angle BAC

and intersects intersects

BC

at point

L

. The chord

DK

is perpendicular to

AC

and intersects it at point

M

. If

\frac{BL}{BC} = 12

, then the ratio

\frac{AM}{MC} =

...
p15. Two dice have numbers

1

through

6

that can be removed from the dice. The twelfth number is removed from the dice and put into a bag. One number is randomly drawn and placed on one of the two dice. After all the numbers are matched, both dice are thrown simultaneously. The probability of the number seven appearing as the sum of the numbers on the top of the second dice is ...
p16. The number of natural numbers

n

such that each point with the coordinates of natural numbers that lies on the line

x+y = n

has a prime number distance from the point center

(0, 0)

is ...
p17. The natural number

n

that satisfies

(-2004)^n-1900^n + 25^n -121^n

and divides

2000

is ...
p18. Ten students sit in a row. All students get up and sit again in the row with the rules that each student can sit back on the the same chair or in the seat next to the old one. How many ways are there that all of the students can sit back in the row ? ...
p19. The largest natural number

n\le 123456

so that there is a natural number x with the property the sum of all digits of

x^2

equals

n

is...
p20. Suppose

ABC

is a triangle and

P

is a point in the triangle. Let points

D, E, F

lie on sides

BC

CA

AB

respectively such that

PD

is perpendicular to

BC

PE

is perpendicular to

CA

, and

PF

is perpendicular to

AB

. If the triangle

DEF

is equilateral and

\angle APB = 70^o

, then

\angle ACB =

Problem Statement