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/c4h2685439p23297220]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. If

y = f(x)

is a function that satisfies the equation

\frac{x}{|x|} +\frac{|y|}{y} = 2y

, then the resulting area of this function is ...
p2. If

n\ge 1

is a natural number, then the least common multiple of

3^n-3

and

9^n + 9

is ...
p3. Given a square

ABCD

, point

P

is inside the square so that

AP = 3

BP = 7

and

DP = 5

. The area of the square

ABCD

is ...
p4. The

n

-th triangular number is the sum of the first

n

natural numbers. Define

T_n

the sum of the first

n

triangular numbers. If

T_n+xT_{n+1}+yT_{n-2} = n

where

x

and

y

are integers, then

x-y =

...
p5. Circles

\omega_1

and

\omega_2

are tangent at point

A

and have a common tangent

\ell

touching

\omega_1

\omega_2

in a row at

B,C

. If

BD

is diameter of circle

\omega_1

with length

2

, and

BC = 3

, the area of triangle

BDC

is ...
p6. The last number of 2014 digits of

\left\lfloor \frac{60^{2014}}{7} \right\rfloor

is...
p7. For OSP preparation, a teacher conducts coaching to the students for one week. Every day, during the construction week, each student sends

5

emails to another student or teacher. At the closing of the show, half of the students got

6

emails, a third of the students got

4

emails and the rest one email each. The Master got

2014

email. If the teacher is allowed to take leave during the construction week , then the amount of leave used is ... days.
Note: When teachers take leave, students continue to study in the classroom independently and only send emails to fellow students.
p8. The sum of all the integers

x

so that

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

is integer is...
p9. If the roots of the quadratic equation

ax^2 + bx + c = 0

are in the interval

[0, 1]

, then the maximum value of

\frac{(2a-b)(a-b)}{a(a- b + c)}

is ...
p10. All

n\le 1000

are such that the number

9 + 99 + 999 + ... + \underbrace{99...9}_{n \,\, digits}

has exactly

n

digits equal to the number

1

are ...
p11. Given a square

ABCD

with a side length of one unit. Suppose, circle with

AD

as diameter, and select point

E

on side

AB

so the line

CE

is tangent to . The area of triangle

BCE

is ...
p12. A school has four class

11

study groups. Each study groups send two students to a meeting. They will sit in a circle with no two students from one study group sitting close together. The number of ways is ...
(Two ways they sit circular is considered equal if one method can be obtained from the other method another with a rotation).
p13. Dono has six cards. Each card is written as a positive integer. For each round, Dono picks

3

cards at random and adds up the three numbers on the cards. After doing

20

the possibility of choosing

3

6

cards, Dono gets a number

16

10

times and the number

18

10

times. The smallest number that exists on the card is...
p14. For real numbers

t

and positive real numbers

a

and

b

satisfy

2a^2-3abt + b^2 = 2a^2 + abt-b^2 = 0.

The value of

t

is...
p15. Let

S(n)

represent the sum of the digits of

n

. For example

S(567) = 5 + 6 + 7 = 18

. How many natural numbers

n

are less than

1000

such that

\frac{S(n)}{S(n + 1)}

is an integer is ...
p16. Given a triangle

ABC

, with sides:

AB = c

BC = a

CA = b = \frac{1}{2}(a+c)

The largest size of

\angle ABC

is...
p17. Inside triangle

ABC

, draw the point

X, Y,Z

with the rule

\angle XBC=\angle ZBA = \frac{\angle ABC}{3}

\angle XCB=\angle YCA = \frac{\angle BCA}{3}

\angle ZAB=\angle YAC = \frac{\angle BAC}{3}

. The length of angle

\angle XYZ

is...
p18. Suppose

0 < \alpha,\beta, \gamma <\frac{\pi}{2}

and

\alpha+\beta+ \gamma =\frac{\pi}{4}

. The number of triples of positive integers

(a, b, c)

\tan \alpha= \frac{1}{a}

\tan \beta= \frac{1}{b}

, and

\tan \gamma = \frac{1}{c}

is...
p19. All consecutive odd number triples

(a, b, c)

with

a < b < c

such that so that

a^2 + b^2 + c^2

is a number with

4

digits such that are all the digits are the same are ...
p20. It is known that a particle in Cartesian coordinates initially lies at the point origin

(0, 0)

. The particle is moving, each step is one unit in the same direction positive

X

-axis, negative

X

-axis direction, positive

Y

-axis direction, or negative Y-axis direction . The number of ways the particle moves so that after the particle moves 9 steps until the point

(2, 3)

is ...

Problem Statement