MathDB

Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2002 [hide=Part A]Part B consists of 5 essay / proof problems, that one is posted [url=https://artofproblemsolving.com/community/c4h2671384p23150576]here
Time: 90 minutes

\bullet

For each problem you have to submit the answer only.

\bullet

Each correct answer is given a value of 1 and the question that is left blank without an answer or an incorrect answer is given a value of 0.
p1. Let

A = (-1)^{-1}

B = (-1)^1

and

C = 1^{-1}

. What is

A + B + C

?
p2. If

y=\frac{x-1}{2x+3}

, write

x

in terms of

y

.
p3. Let

S = (x-2)^4 + 8(x-2)^3 + 24(x-2)^2 + 32(x-2) + 16

. What is

S

when written in as few terms as possible?
p4. The real number

2,525252...

is a rational number, so it can be written in the form

\frac{m}{n}

, where

m, n

are integers,

n \ne 0

. If

m

and

n

are relatively prime, what is

m + n

?
p5. Suppose

M

and

m

represent the largest and smallest numbers, respectively, of all

4

-digit numbers whose four digits add up to

9

. What is the largest prime factor of

M-m

?
p6. Consider an equation of the form

x^2 + bx + c = 0

. How many such equations have real roots if the coefficients

b

and

c

can only be chosen from the set

\{1, 2, 3, 4, 5, 6\}

?
p7. Given three numbers

k

m

and

n

. The statement “If

k\ge m

, then

k > n

” is not true. Is the statement correct in this case?
p8. A drain should be constructed using a

10

cm diameter pipe. However, only small pipes with a diameter of

3

cm are available. So that the channel capacity is not smaller than desired, how many

3

cm pipes need to be used instead of one

10

cm pipe?
p9. An equilateral triangle, a circle and a square have the same perimeter. Of the three shapes, which one has the largest area?
p10. Triangle

ABC

has side lengths

AB = 10

BC = 7

, and

CA = 12

. If each side is extended to three times its original length, then the triangle formed has an area of how many times the area of

​​ABC

?
p11. A total of

n

board members of an organization shall be divided into four commissions according to the following provisions: (i) each member belongs to exactly two commissions, and (ii) each two commissions has exactly one joint member. What is

n

?
p12. Define

a*b = a + b + ab

, for all real numbers

a, b

. If

S = \{a

real number such that

a*(-a) > a\}

, write

S

as an interval.
p13. The diameter of a semicircle coincides with the base

AB

of triangle

ABC

. The vertex

C

moves in such a way that the midpoint of side

AC

always lies on the semicircle. What is the curvature of the locus of point

C

?
p14. What is the largest positive integer that divides all the numbers

1^5-1, 2^5-2,..., n^5-n,...

?
p15. If

2002 = a_1 + a_2 \cdot 2! + a_3 \cdot 3! + ...+ a_n \cdot n!

, where

a_k

is an integer,

0 \le a_k \le k

k = 1, 2, ... , n

, and

a_n \ne 0

, find the ordered pair

(n, a_n)

.
p16. What is the remainder of the division of

43^{43^{43}}

100

?
p17. Four couples buy tickets for

8

seats in a row at a show. Two people will sit next to each other only if they are both married or of the same sex. How many ways to put the four couples in the

8

seats?
p18. How many

4

-digit numbers are in the form of

\overline{abcd}

with

a \le b \le c \le d

?
p19. We draw a regular polygon

R

with

2002

vertices and all its diagonals. How many triangles are formed where all the vertices are vertices of

R

, but none of the sides are sides of

R

?
p20. A marathon race followed by four SMUs: Peacock, Pigeon, Sparrow and Swallow. Each high school sent five runners. Runners who enter the

1

st,

2

nd,

3

rd,

4

th,

5

th,

6

th finishes obtain consecutive values of

7, 5, 4, 3, 2, 1

. The value of each SMU is the sum of the values of the fifth runner. The SMU with the highest value is the champion of the race. At the end of the race, SMU Pipit became the champion and no two runners finished at the same time. How many possible SMU values are there?

Problem Statement