MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2016 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685861p23301381]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. Suppose

a, b, c

are three natural numbers that satisfy

2^a + 2^b + 2^c = 100

. The value of

a + b + c

is ... .
p2. A function f has the property

f(65x+1) = x^2-x+1

for all real numbers

x

. The value of

f (2016)

is ... .
p3. Three different numbers

a, b, c

will be chosen one by one at random from

1, 2, 3, 4,..., 10

with respect to the order. The probability that

ab+c

is even is ...
p4. Point

P

is a point on the convex quadrilateral

ABCD

with

PA = 2

PB = 3

PC = 5

, and

PD = 6

. The maximum area of the quadrilateral

ABCD

is...
p5. If

0 < x <\frac{\pi}{2}

and

4 \tan x + 9 \cot x \le 12

, then the possible values of

\sin x

are ... .
p6. For every natural number

n

, let

s(n)

represent the result of the sum of the digits of n in decimal writing. For example,

s(2016) = 2+0+1+6 = 9

. The result of the sum of all natural numbers n such that

n + s(n) = 2016

is ...
p7. Among

30

students,

15

students liked athletics,

17

students liked basketball, and

17

students love chess. Students who enjoy athletics and basketball as much as students who love basketball and chess. A total of

8

students enjoy athletics and chess. Students who like basketball and chess are twice as much as students who love happy all three. Meanwhile,

4

students were not happy with any of the three. From the

30

students, three students were chosen randomly. Probability of each each student who is selected only likes chess or basketball is ...
p8. Given a cube

ABCD:EFGH

with side length

5

. Points

I

and

J

any point on

BF

with

IJ = 1

. Any point

K

and

L

CG

with

KL = 2

. Ants move from

A

H

on the path

AIJKLH

. The shortest path is...
p9. The number of triple prime numbers

(p, q, r)

that satisfies

15p+7pq+qr = pqr

is ...
p10. If

x^2 + xy + 8x =-9

and

4y^2 + 3xy + 16y =-7

, then the value of

x + 2y

maybe is ...
p11. The lengths of the sides of a triangular pyramid are all integers. Its five ribs are

14

long each;

20

40

52

, and

70

. The number of possible lengths of the sixth side is...
p12. A chess player plays a minimum of once every day for seven day with a total of

m

matches. Maximum value of

m

such that there are two or more consecutive days with a total of four matches is ...
p13. Mr. Adi's house has a broken water meter, that cannot show numbers

3

and

9

. For example, the number shown on the meter after

22

24

and also the number shown after

28

40

. For example, in one month, Pak Adi's water meter shows

478

^3

. The actual loss of Mr. Adi because the broken meter is ... m

^3

.
p14. The result of the sum of all the real numbers

x

that fulfil

\lfloor 8x-1008 \rfloor + \lfloor x \rfloor = 2016

is ... .
p15. Suppose

a_1, a_2, ..., a_{120}

are

120

permutations of the word MEDAN which is sort alphabetically like in a dictionary, for example

a_1 = ADEMN

a_2 = ADENM

a_3 = ADMEN

, and so on. The result of the sum of all

k

indexes until the letter

A

is the third letter in the permutation

a_k

is ...
p16. Suppose

ABCDE

is a regular pentagon with area

2

. The points

P, Q, R, S, T

are the intersection of the diagonals of the pentagon

ABCDE

such that

PQRST

is a regular pentagon. If the area of

PQRST

is written in the form

a-\sqrt{b}

with

a

and

b

natural numbers, then the value of

a + b

is ...
p17. Triangle

ABC

has a circumcircle of radius

1

. If the two medians of triangle ABC each have a length of

1

, then the perimeter of the triangle

ABC

is ....
p18. Sequence

x_0, x_1, x_2, ... , x_n

is defined as

x_0 = 10

_x1 = 5

, and

x_{k+1} = x_{k-1}-\frac{1}{x_k}

for

k = 1, 2, 3, ..., n-1

and we get

x_n = 0

. The value of

n

is ...
p19. In a soccer tournament involving

n

teams, each team plays against the other team exactly once. In one game,

3

points will be awarded to the winning team and

0

points to the losing team, while

1

point is given to each team when the match ends in a draw. After the match ends, only one team gets the most points and only that team gets the number of wins at least. The smallest value of

n

so that this is possible ...
p20. The sequence of non-negative numbers

a_1, a_2, a_3, ...

is defined with

a_1 = 1001

and

an+2 = |a_{n+1}-a_n|

for

n\ge 1

. If it is known that a_2 < 1001 and

a_{2016} = 1

, so the number of possible values of

a_2

is...

Problem Statement