MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2018 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2686089p23303313]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. The number of ordered pairs of integers

(a, b)

so that

a^2 + b^2 = a + b

is ..
p2. Given a trapezoid

ABCD

, with

AD

parallel to

BC

. It is known that

BD = 1

\angle DBA = 23^o

, and

\angle BDC = 46^o

. If the ratio

BC: AD = 9:5

, then the length of the side

CD

is ...
p3. Suppose

a > 0

and

0 < r_1 < r_2 < 1

so that

a + ar_1 + ar_1^2 + ...

and

a + ar_2 + ar_2^2+...

are two infinite geometric series with the sum of

r_1

and

r_2

, respectively. The value of

r_1 + r_2

is ....
p4. It is known that

S = \{10,11,12,...,N\}

. An element in

S

is said to be trubus if the sum of its digits is the cube of a natural number. If

S

has exactly

12

trubus, then the largest possible value of

N

is....
p5. The smallest natural number

n

such that

\frac{(2n)!}{(n!)^2}

to be completely divided by

30

is ..
p6. Given an isosceles triangle

ABC

with

M

midpoint of

BC

. Let

K

be the centroid of triangle

ABM

. Point

N

lies on the side

AC

so that the area of the quadrilateral

KMCN

is half of the area of triangle

ABC

. The value of

\frac{AN}{NC}

is ...
p7. In a box there are

n

red marbles and

m

blue marbles. Take

5

marbles at a time. If the probability that

3

red marbles and

2

blue marbles are drawn is

\frac{25}{77}

, then the smallest possible value of

m^2 + n^2

is ...
p8. Let P(x) be a non-constant polynomial with a non-negative integer coefficient that satisfies

P(10) = 2018

. Let

m

and

M

be the minimum and maximum possible values of

P(l)

, respectively). The value of

m + M

is ...
p9. A province consists of nine cities named

1, 2, 3, 4, 5, 6, 7, 8, 9

. From city

a

there is a direct road to city

b

if and only if

\overline{ab}

and

\overline{ba}

are two-digit numbers that are divisible by

3

. Two distinct cities

a_1

and

a_n

are said to be connected if there is a sequence of cities

a_1,a_2,...,a_{n-1},a_n

, so that there is a direct path from

a_i

a_{i+ i}

for each

i =1, ... , n -1

. The number of cities connected to city

4

is ...
p10. Given

37

points as shown in the figure so that every two neighboring points are one unit apart. From each of the three distinct points a red triangle is drawn. The number of possible sidelengths of an equilateral red triangle is ... https://cdn.artofproblemsolving.com/attachments/e/1/75f1cb9051226e09fc52673762276564a32210.png
p11. A positive integer

k

is taken at random with

k \le 2018

. The probability that

k^{1009}

has a remainder of

2

when divided by

2018

is ....
p12. Given the non-negative real numbers

a, b, c, d, e

where

ab + bc + cd + de = 2018

. The minimum value of

a + b + c + d + e

is ...
p13. The number of subsets (including empty sets) of

X = \{1,2,3,... ,2017,2018\}

which does not have two elements

x

and

y

so that

xy = 2018

there are

m2^n

with

m

odd. The value of

m + n

is ...
p14. Let

S = \{ 1 ,2 ,..., n\}

. It is known that there are exactly

1001

pairs

(a, b, c, d)

with

a, b, c, d \in S

and

a < b < c < d

so that

a, b, c, d

are arithmetic sequences. The value of

n

is ....
p15. The number of natural numbers

n

such that

n^4 - 5n^3- 5n^2 + 4n + 10

is a prime number is...
p16. Point

M

lies on the circumcircle of regular pentagon

ABCDE

. The greatest value that

\frac{MB + ME}{MA + MC + MD}

might be is ...
p17. For

x, y

non-zero real numbers, the sum of the maximum and minimum values of

\frac{xy - 4y^2}{x^2 + 4y^2}

is ...
p18. An alien race has a unique language consisting of only two letters

X

and

Z

. In this language, each word consists of at least one letter and no more than

11

letters. For every two words, if the first and second words are written side by side then the result is not a word. For example, if

XXZ

and

ZZZZX

are words, then

XXZZZZZX

is not a word. The maximum number of words in this language is ...
19. An acute triangle

ABC

has integer sidelengths. It is known that

AC = BD

where

D

is a point on the line

BC

such that

AD

is perpendicular to

BC

. The smallest possible length of side

BC

is ...
p20. The largest natural number

n

such that

50\lfloor x \rfloor - \lfloor x \lfloor x \rfloor \rfloor = 100n - 27 \lceil x \rceil

has a real solution

x

is ...

Problem Statement