MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2004 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c4h2671387p23150599]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. Let

x

and

y

be non-zero real numbers. If

\frac{1}{x}+\frac{1}{y}=10

and

x + y = 40

, what is

xy

?
p2. A bottle of syrup can be used to make

60

glasses of drink if dissolved in water with a ratio of

1

part syrup to 4 parts water. How many glasses of drink are obtained from a bottle of syrup if the ratio of the solution is

1

part syrup to

5

parts water?
p3. The population of Central Java is

25\%

of the population of the island of Java and

15\%

of the population of Indonesia. What percentage of Indonesia's population lives outside Java?
p4. When calculating the volume of a cylinder, Dina made a mistake. He entered the diameter of the base into the formula for the volume of the cylinder, when he should have entered the radius of the base. What is the ratio of the results of the Office's calculations to the expected results?
p5. Three circles through the center of the coordinates

(0, 0)

. The center of the first circle is in quadrant I, the center of the second circle is in quadrant II and the center of the third circle is in quadrant III. If P is a point inside the three circles, which quadrant is this point in?
p6. Given successively (from left to right) the first, second and third images of a row of images. How many black circles are in the nth picture? https://cdn.artofproblemsolving.com/attachments/6/f/2ef5412d46111bca64b9faeea5eab54550c4f1.png
p7. Given a triangle ABC with side length ratio

AC : CB = 3: 4

. The bisector of the exterior angle

C

intersects the extension of

BA

P

(point A lies between the points

P

and

B

). Determine the ratio of the length of

PA: AB

.
p8. How many sequences of non-negative integers

(x, y, z)

satisfy the equation

x + y + z = 99

?
p9. Find the set of all natural numbers n such that

n(n-1)(2n-1)

is divisible by

6

.
p10. Find all real numbers x that satisfy

x^2 < |2x-8|

.
p11. From between

6

cards numbered

1

6

two cards are drawn at random. What is the probability of drawing two cards whose sum is

6

?
p12. In a trapezoid of height

4

, the two diagonals are perpendicular to each other. If one of the diagonals is

5

, what is the area of the trapezoid?
p13. Determine the value of

\left(1-\frac23\right)\left(1-\frac25\right)\left(1-\frac27\right)...\left(1-\frac{2}{2005}\right)

.
p14. Santi and Tini run along a circular track. Both start running at the same time from point

P

, but take opposite directions. Santi runs

1\frac12

times faster than Tini. If

PQ

is the center line of the circular path and the two meet for the first time at point

R

, what is the measure of

\angle RPQ

?
p15. On the sides

SU

TS

and

UT

of triangle

STU

, the points

P, Q

and

R

are selected so that

SP = \frac14 SU

TQ = \frac12 TS

and

UR = \frac13 UT

. If the area of triangle

STU

1

, what is the area of triangle

PQR

?
p16. Two real numbers

x, y

satisfy

\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1

. What is the value of

x + y

?
p17. What is the minimum number of points that must be taken from a square with a side length of

2

, so that it is guaranteed to always pick two points whose distance between them is not more than

\frac12 \sqrt2

?
p18. Let

f

be a function that satisfies

f(x)f(y)-f(xy) = x + y

, for every integer

x

and

y

. What is the value of

f(2004)

?
p19. The notation

gcd \, (a, b)

expresses the greatest common divisor of the integers

a

and

b

. Three natural numbers

a_1 < a_2 < a_3

satisfy

gcd \,(a_1, a_2, a_3) = 1

, but

gcd \,(a_i, a_j) > 1

i\ne j

i, j = 1, 2, 3

. Find

(a_1, a_2, a_3)

so that

a_1 + a_2 + a_3

is minimal.
p20. Define

a * b = a + b + ab

, for all integers

a, b

. We say that the integer

a

is a factor of the integer

c

if there is an integer

b

that satisfies

a * b = c

. Find all the positive factors of

67

Problem Statement