MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2015 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685462p23297434]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 sum of all the real numbers x that satisfy

x^2-2x = 2 + x\sqrt{x^2-4x}

is ...
p2. The number of integers

n

, so that

n + 1

is a factor of

n^2 + 1

is ...
p3. At a party, each man shakes hands with another man only once. Likewise, each woman only shakes hands once with another woman who attended the party. No shaking hands between men and the women at the party. If many men are present at the party more than women and the number of handshakes between men or women is there

7

handshakes. The number of men present at the party is...
p4. Given a triangle

ABC

, through the point

D

which lies on the side

BC

are drawn the lines

DE

and

DF

, parallel to

AB

and

AC

, respectively, (

E

AC

F

AB

). If the area of triangle

DEC

4

times the area of triangle

BDF

, then the ratio of the area of triangle

AEF

to the area of triangle

ABC

is ...
p5. If

f

is a function defined on the set of real numbers, such that

3f(x)-2f(2-x) = x^2 + 8x-9

for all real numbers

x

, then the value of

f(2015)

is ...
p6. The number of pairs of integers

(a, b)

that satisfy

\frac{1}{a}+\frac{1}{b + 1}=\frac{1}{2015}

is ...
p7. There are

10

people, five boys and five girls, including a couple of bride. A photographer who is not one of the

10

people will take pictures of six of them, including the two brides, with neither two men nor two women close together. The number of ways is ...
p8. The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The value of the cosine of the smallest angle is ...
p9. Given two different squared terms

f(x) = x^2 + ax + b

and

g(x) = x^2 + cx + d

which satisfies

f(20) + f(15) = g(20) + g(15)

. The sum of all real numbers

x

that satisfy

f(x) = g(x)

is equal to ...
p10. Given a and b positive integers with

\frac{53}{201} <\frac{a}{b}<\frac{4}{15}

. The smallest possible value of

b

is ...
p11. Suppose in a laboratory there are

20

computers and

15

printers. Cables are used to connect computers and printers. Sadly, one printer can only serve one computer at a time together. Desired

15

computers can always use the printer on same time. The number of cables required to connect at least as many computers and printers is ...
p12. Given a triangle

ABC

with

M

in the midpoint of

BC

, and on the side AB selected point

N

so that

NB = 2NA

. If

\angle CAB = \angle CMN

, then the value of

\frac{AC}{BC}

is ...
p13. Given the sequence

a_0, a_1, a_2, ...

with

a_0 = 2

a_1 =\frac83

and

a_ma_n= a_{m+n}- a_{m-n}

for every natural number

m, n

with

m\ge n

. The number of natural numbers

n

that fulfills

a_n-3^n > \frac{1}{2015}

is ...
p14. For the real number x, the notation

\lfloor x \rfloor

denotes the largest integer that not greater than

x

, while

\lceil x \rceil

represents the smallest integer which is not smaller than

x

. The real number

x

that satisfies

\lfloor x^2 \rfloor -3x + \lceil x \rceil = 0

is ...
p15. A circle intersects an equilateral triangle

ABC

at six points that are different. About the six intersection points, every two points located on a different side of the triangle, so that:

B,D,E,C

and

C, F, G,A

, and

A,H,J,B

are in a row in a line. If

AG = 2

GF = 13

FC = 1

, and

HJ = 7

, then the length of

DE

is ...
p16. In the picture there are as many triangles as ...... https://cdn.artofproblemsolving.com/attachments/b/b/bfd55c68a906f7b4c41ffa07728f0602f2afc1.png
p17. Let

M

and

m

be the largest and smallest values of a, respectively such that

\left|x^2-2ax-a^2-\frac34 \right| \le 1

for every

x\in[0, 1]

. The value of

M-m

is ...
p18. All integers

n

such that

\frac{9n + 1}{n + 3}

is the square of a rational number are ...
p19. The set

A

, subset of

\{1, 2,..., 15\}

is said to be good, if for every

a \in A

applies

a-1 \in A

a + 1 \in A

. The number of good subsets with five elements of of

\{1, 2,..., 15\}

is ...
p20. Given an isosceles triangle

ABC

, where

AB = AC = b

BC = a

, and

\angle BAC = 100^o

. If

BL

bisects

\angle ABC

, then the value of

AL + BL

is ...

Problem Statement