MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2006 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684669p23288980]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.
p1. The sum of all integers between

\sqrt[3]{2006}

and

\sqrt{2006}

is ...
p2. In trapezoid

ABCD

, side

AB

is parallel to

DC

. A circle tangent to all four sides of the trapezoid can be drawn. If

AB = 75

and

DC = 40

, then the perimeter of the trapezoid

ABCD = ...

p3. The set of all

x

that satisfies

(x-1)^3 + (x-2)^2 = 1

is ...
p4. The largest two-digit prime number which is the sum of two other prime numbers is ...
p5. Afkar selects the terms of an infinite geometric sequence

1, \frac12, \frac14, \frac18

, to create a new infinite geometric sequence whose sum is

\frac17

. The first three terms Afkar chooses are ...
p6. The area of the sides of a cuboid are

486

486

243

243

162

162

. The volume of the cuboid is ...
p7. The maximum value of the function

f(x) = \left(\frac13 \right)^{x^2-4x+3}

is ...
p8. Given the function

f(x) = ||x - 2| - a| -3

. If the graph f intersects the

x

-axis at exactly three points, then

a =. ..

p9. For natural numbers

n

, write

s(n) = 1 + 2 + ...+ n

and

p(n) = 1 \times 2 \times ... \times n

. The smallest even number

n

such that

p(n)

is divisible by

s(n)

is ...
p10. If

|x|+ x + y = 10

and

x + |y|-y = 12

, then

x + y = ...

p11. A set of three natural numbers is called an arithmetic set if one of its elements is the average of the other two elements. The number of arithmetic subsets of

\{1,2,3,...,8\}

is ...
p12. From each one-digit number

a

, the number

N

is made by juxtaposing the three numbers

a + 2

a + 1

a

i.e.

N = \overline{(a+2)(a+1)a}

. For example, for

a = 8

N = 1098

. The ten such numbers

N

have the greatest common divisor ...
p13. If

x^2+\frac{1}{x^2}=47

, then

\sqrt{x}+\frac{1}{\sqrt{x}}= ...

p14. A class will choose a student from among them to represent the class. Every student has the same opportunity to be selected. The probability that a male student is selected is

\frac23

times the probability that a female student is selected. The percentage of male students in the class is ...
p15. In triangle

ABC

, the bisector of angle

A

intersects side

BC

at point D. If

AB = AD = 2

and

BD = 1

, then

CD = ...

p16. If

(x-1)^2

divides

ax^4 + bx^3 + 1

, then

ab = ...

p17. From point

O

, two half-lines (rays)

l_1

and

l_2

are drawn which form an acute angle . The different points

A_1, A_3, A_5

lie on the line

l_2

, while the points

A_2, A_4, A_6

lie on the line

l_1

. If

A_1A_2 = A_2A_3 = A_3A_4 = A_4O = OA_5 = A_5A_6 = A_6A_1

, then = ...
p18. The number of different

7

-digit numbers that can be formed by changing the order of the numbers

2504224

is ...
p19. Evan creates a sequence of natural numbers

a_1, a_2, a_3

, which satisfies

a_{k+1}-a_k = 2(a_k-a_{k-1})-1

, for

k = 2, 3,...,

and

a_2 - a_1 = 2

. If

2006

appears in the sequence, the smallest possible value of

a_1

is ...
p20. In triangle

ABC

, the medians from vertex

B

and vertex

C

intersect at right angles to each other. The minimum value of

\cot B + \cot C

is ...

Problem Statement