MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2010 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685201p23295229]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. Calculate

\sum_{j=0}^{n}\left( {n \choose j}\left( \sum_{i=0}^{n} {j \choose i}8^i\right)\right)

p2. In triangle ABC, let

a

b

, and

c

be the side lengths of

BC

CA

, and

AB

, respectively. If

\frac{2a}{\tan A}=\frac{b}{\tan B}

then the value of

\frac{\sin^2 A-\sin^2 B}{\cos^2 A+\cos^2 B}

is ...
p3. Given a polynomial

P(x) = x^4 + ax^3 + bx^2 + cx + d

with

a, b, c

, and

d

constants. If

P(1) = 10

P(2) = 20

, and

P(3) = 30

, then the value of

\frac{P(12)+P(-8)}{10}

is ...
p4. Let

S = \{1, 2, 3, 4, 5\}

. The number of functions

f : S\to S

that satisfies f(f(x)) = x for all

x\in S

is ..
p5. If

a, b

, and

c

represent the lengths of the sides of a triangle that satisfies

(a + b + c)(a + b-c) = 3ab

, then the measure of the angle opposite the side of length

c

is ...
p6. The number of six-digit numberz

\overline{abcdef}

with

a > b > c\ge d > e > f

is ...
p7. The prime number

p

so that

p^2 + 73

is a perfect cube is ...
p8. Given triangle

ABC

is right-angled at

C

AC = 3

, and

BC = 4

. Triangle

ABD

is right-angled at

A

AD = 12

and points

C

and

D

are opposite to side

AB

. Parallel line

AC

through

D

cut the extension of

CB

E

. If

\frac{DE}{DB}=\frac{m}{n}

where

m

and

n

are relatively prime positive integers, then

m + n = ...

p9. On a circle there are

12

distinct points. By using these

12

points we will make

6

non-intersecting chord. There are ... ways to do it.
p10. The number of members of the set

S = \{gcd (n^3 + 1, n^2 + 3n + 9)|n\in Z\}

is ...
p11. The quadratic equation

x^2-px-2p = 0

has two real roots

a

and

b

. If

a^3 + b^3 = 16

, then the sum of satisfying values of

p

is ...
p12. In a plane, there are

n

points with coordinates a pair of integers. The smallest value of

n

so that there are two points whose midpoints also have both coordinate integer pairs is ...
p13. The natural number

n

such that the equation

x\left[\frac{1}{x}\right]+\frac{1}{x} \left[x\right]=\frac{n}{n+1}

has exactly

2010

positive real solution is ⋅⋅⋅⋅⋅⋅
Note: For any real number

x

is defined

[a]

as the largest integer less than or equal to with

x

.
p14. Two circles (not equally large) intersect on the outside. Points

A

and

A_1

are located on the small circle, while

B

and

B_1

are on the large circle. The lines

PAB

and

PA_1B_1

, are common tangent lines of the two circles. If

PA = AB = 4

, then the area of the small circle is ...
p15. Twenty -seven students in a class will be made into six discussion groups each consisting of four or five σtudents. The number of ways is ...
p16. Someone wrote a chain letter to

6

people. The recipient of this letter is instructed to sent letters to

6

other people. All recipients of the letter read the contents of the letter and then some people carry out the orders written in the letter, the rest do not continue the chain letter this. If there are

366

people who do not continue this chain letter, then the number of people that resides in this chain mail system is ...
p17. The sum of the constant terms of

\left(x^5 - \frac{2}{x^3}\right)^8

is ...
p18. The number of positive integers

n < 100

, so the equation

\frac{3xy-1}{x+y}=n

has a solution pair of integers

(x, y)

is ...
p19. It is known that

x, y

, and

z

are real numbers that satisfy the system of equations:

x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}

xyz = 1

The smallest value of

|x + y + z|

is ...
p20. Triangle

ABC

has side lengths

BC = 5

AC = 12

, and

AB = 13

. Point

D

is on

AB

and point

E

AC

. If

DE

divides triangle ABC into two equal parts, then the minimum length of

DE

is ...

Problem Statement