MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2007 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684679p23289059]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 largest

4

-digit odd number who is sum of all prime numbers is ...
p2. The amount of money consists of coins in the

500

Rp.,

200

Rp. , and

100

Rp. with a total value of

100,000

Rp.. If the

500

's note is half the

200

's note, but three times the

100

's note, then the number of coins is ...
p3. The length of the hypotenuse of a right triangle is equal to twice the length of the shortest side, while the length of the third side is

1

unit longer than the length of the shortest side. The area of the triangle is ... square units .
p4. Among the numbers

2006

2007

and

2008

, the number that has the most different prime factors is ...
p5. A used car dealer sells two cars for the same price. He loses

10\%

on the first car, but breaks even (return on capital) for both cars. The merchant's profit percentage for the second car is
p6. Dona arranges five congruent squares into a flat shape. No square overlaps another square. If the area of the figure obtained by Dona is

245

^2

, the perimeter of the figure is at least ... cm.
p7. Four football teams enter a tournament. Each team plays against each of the other teams once. Each time a team competes, a team gets

3

points if it wins,

0

if it loses and

1

if the match ends in a draw. At the end of the tournament one of the teams gets a total score of

4

. The sum of the total points of the other three teams is at least ...
p8. For natural numbers

n

. In its simplest form,

1!1+2!2+3!3+...+n!n=...

p9. Point

P

is located in quadrant I on the line

y = x

. The point

Q

lies on the line

y = 2x

such that

PQ

is perpendicular to the line

y = x

and

PQ = 2

. Then the coordinates of

Q

are ...
p10. The set of all natural numbers

n

such that

6n + 30

is a multiple of

2n + 1

is ...
p11. The term constant in the

\left(2x^2-\frac{1}{x}\right)^9

expansion is ...
p12. The abscissa of the point where the line

\ell

intersects the

x

-axis and the coordinates of the point where it intersects

\ell

with the

y

-axis are prime numbers. If

\ell

also passes through the point

(3, 4)

, the equation of

\ell

is ...
p13. Seventeen candies are packed into bags so that the number of candies in each two bags is at most

1

. The number of ways to pack the candies into at least two bags is ...
p14. If the maximum value of

x + y

in the set

\{(x, y)|x\ge 0, y\ge0, x + 3y \le 6, 3x + y \le a\}

4

, then

a = ...

p15. A

5 \times 5 \times 5

cube is composed of

125

unit cubes. The surface of the large cube is then painted. The ratio of the sides (surfaces) of the

125

painted to unpainted unit cubes is ....
p16. A square board is divided into

4 \times 4

squares and colored like a chessboard. Each tile is numbered from

1

16

. Andi wants to cover the tiles on the board with

7

cards of

2 \times 1

tile. In order for his

7

cards to cover the board, he must discard two squares. The number of ways he disposes of two squares is ...
p17. The natural numbers

1, 2, ... , n

are written on the blackboard, then one of the numbers is deleted. The arithmetic mean of the numbers left behind is

35 \frac{7}{17}

. The

n

-th possible number for this to happen is ...
p18. Given a triangle

ABC

right at

A

, point

D

AC

and point

F

BC

. If

AF\perp BC

and

BD = DC = FC = 1

, then

AC = ...

p19. Among all the natural number solutions

(x, y)

of the equation

\frac{x+y}{2}+\sqrt{xy}=54

, the solution with the largest

x

(x, y) = ...

p20. Let

V

be the set of points on the plane with integer coordinates and

X

be the set of midpoints of all pairs of points in the set

V

. To ensure that any member of

X

also has integer coordinates, the number of members of

V

must be at least ...

Problem Statement