MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2017 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685870p23301432]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. Two nonzero real numbers

a

and

b

satisfy

ab = a - b

. The possible value of

\frac{a}{b}+\frac{b}{a}- ab

is ...
p2. Community leaders somewhere in RW, apart from Mr. RW and Mrs. RW, there are

5

women and

6

men. Kelurahan asked

6

people to attend a seminar at the city level.

6

people were chosen as RW delegates, with a composition of

3

women and

3

men, one of whom was Mr. RW. The number of ways to choose the delegate is ...
p3. Given a triangle

ABC

with

AB = 13

AC = 15

, and the length of the altitude on BC is

12

. The sum of all possible lengths of

BC

is ...
p4. The two-digit prime number

p =\overline{ab}

that satisfies

\overline{ba}

is also prime is ...
p5. Suppose

f

is a real function that satisfies

f \left(\frac{x}{3}\right) = x^2 +2x+3

. The sum of all

z

values that satisfy

f(3z) = 12

is ...
p6. Ita chooses 5 numbers from

\{1,2,3,4,5, 6, 7\}

and tells Budi the product of the five numbers. Then Ita asked if Budi knew that the sum of the five numbers was an odd or even number. Budi replied that he couldn't be sure. The value of the product of five numbers owned by Ita is ...
p7. Let

ABCD

be a square with side length

2017

. Point

E

lies on the segment

CD

CEFG

is a square with side length

1702

F

and

G

lie outside

ABCD

. If the circumcircle of the triangle

ACF

intersects

BC

again at point

H

, then the length of

CH

is ...
p8. The number of pairs of natural numbers

(x, y)

that satisfy the equation

x + y = \sqrt{x} + \sqrt{y} + \sqrt{xy}

is ...
p9. Let

x

and

y

be real numbers that satisfy the equation

x^2y^2 + 4x^2 + y^2 + 1 = 6xy

. If

M

and

m

represent the largest and smallest possible values of

x - y

, respectively, then the value of

M - m

is ...
p10. Given a 2017 lamp equipped with a switch to turn the lights on and off. At first all the lights were off. Every minute Ani has to press exactly 5 switches. Every time the switch is pressed, the light that was extinguished becomes on and the light that was lit becomes extinguished. To turn on all the lights Ani requires at least ... minutes.
p11. Given a positive real number

k

. In a triangle

ABC

the points

D, E

, and

F

lie on sides

BC, CA

, and

AB

respectively so that

\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}=k

[ABC]

and

[DEF]

represent the area of triangles

ABC

and

DEF

, respectively, then

\frac{[ABC]}{[DEF]}=

...
p12. For any natural number

k

, let

I_k= 10... 064

with

k

times

0

between 1 and

6

. If

N(k)

represents the number of factors of

2

in the prime factorization of

I_k

, then the maximum value for

N(k)

is ...
p13. If

x, y

, and

z

are positive real numbers that satisfy

x+\frac{1}{y}=4, y+\frac{1}{z}=1, z+\frac{1}{z}=\frac{7}{3}

then the value of

xyz

is ...
p14. Ten students have different heights. The gym teacher wanted them to line up sideways, on the condition that no student was flanked by two other students who were taller than him. The number of ways to make such a sequence is ...
p15. Given a triangle

ABC

with

\omega

as the outer circle. The chord

AD

is the bisector of the angle

BAC

that intersects

BC

at point

L

. The chord

DK

is perpendicular to

AC

and intersects

AC

at point

M

. If

\frac{BL}{LC}=\frac{1}{2}

then the value of

\frac{AM}{MC}=

is.....
p16. The original four-digit number

n

is completely divided by

7

. The original number

k

, obtained by writing the n-digits from back to front, is also completely divided by

7

. In addition, it is known that

n

and

k

have the same remainder when divided by

37

. If

k> n

, then the sum of all

n

that satisfy is .....
p17. Given

\{a_i\}_{i\ge 1}

real numbers with

a_1 = 20,17

. If

a_1,a_2,...,a_{11}

and

\lfloor a_1 \rfloor , \lfloor a_2\rfloor , ..., \lfloor a_{10} \rfloor

, are each an arithmetic sequence; whereas

\lfloor a_1 \rfloor , \lfloor a_2\rfloor , ..., \lfloor a_{11} \rfloor

is not arithmetic sequence, then the minimum value of

a_2 - a_1 -\lfloor a_2-a_1 \rfloor

is ...
p18. In a Snack Center there are four shops each selling three type of food. There are

n

people who each buy exactly one food at each store. For every three shoppers there is at least one store where all three types of food are bought. The maximum possible value of

n

is ...
19. Given is the regular hexagon

ABCDEFG

. The distances from

A

on the lines

BC

BE

CF

, and

EF

respectively are

a, b, c

, and

d

. The value of

\frac{ad}{bc}

is ...
20. It is known

f (x)

is a polynomial of degree

n

with integer coefficients satisfying

f(0) = 39, \,\,\, f(x_1) = f(x_2) = f(x_3) = ... = f (x_n) = 2017

, with

x_1, x_2, x_3, ..., x_n

are all different. The largest possible number of

n

is .....

Problem Statement