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2002 Indonesia Regional
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Indonesia Regional MO 2002 Part A 20 problems 90' , answer only
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
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For each problem you have to submit the answer only.
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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
A = (-1)^{-1}
A
=
(
−
1
)
−
1
,
B
=
(
−
1
)
1
B = (-1)^1
B
=
(
−
1
)
1
and
C
=
1
−
1
C = 1^{-1}
C
=
1
−
1
. What is
A
+
B
+
C
A + B + C
A
+
B
+
C
?p2. If
y
=
x
−
1
2
x
+
3
y=\frac{x-1}{2x+3}
y
=
2
x
+
3
x
−
1
, write
x
x
x
in terms of
y
y
y
.p3. Let
S
=
(
x
−
2
)
4
+
8
(
x
−
2
)
3
+
24
(
x
−
2
)
2
+
32
(
x
−
2
)
+
16
S = (x-2)^4 + 8(x-2)^3 + 24(x-2)^2 + 32(x-2) + 16
S
=
(
x
−
2
)
4
+
8
(
x
−
2
)
3
+
24
(
x
−
2
)
2
+
32
(
x
−
2
)
+
16
. What is
S
S
S
when written in as few terms as possible?p4. The real number
2
,
525252...
2,525252...
2
,
525252...
is a rational number, so it can be written in the form
m
n
\frac{m}{n}
n
m
, where
m
,
n
m, n
m
,
n
are integers,
n
≠
0
n \ne 0
n
=
0
. If
m
m
m
and
n
n
n
are relatively prime, what is
m
+
n
m + n
m
+
n
?p5. Suppose
M
M
M
and
m
m
m
represent the largest and smallest numbers, respectively, of all
4
4
4
-digit numbers whose four digits add up to
9
9
9
. What is the largest prime factor of
M
−
m
M-m
M
−
m
?p6. Consider an equation of the form
x
2
+
b
x
+
c
=
0
x^2 + bx + c = 0
x
2
+
b
x
+
c
=
0
. How many such equations have real roots if the coefficients
b
b
b
and
c
c
c
can only be chosen from the set
{
1
,
2
,
3
,
4
,
5
,
6
}
\{1, 2, 3, 4, 5, 6\}
{
1
,
2
,
3
,
4
,
5
,
6
}
?p7. Given three numbers
k
k
k
,
m
m
m
and
n
n
n
. The statement “If
k
≥
m
k\ge m
k
≥
m
, then
k
>
n
k > n
k
>
n
” is not true. Is the statement correct in this case?p8. A drain should be constructed using a
10
10
10
cm diameter pipe. However, only small pipes with a diameter of
3
3
3
cm are available. So that the channel capacity is not smaller than desired, how many
3
3
3
cm pipes need to be used instead of one
10
10
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
A
B
C
ABC
A
BC
has side lengths
A
B
=
10
AB = 10
A
B
=
10
,
B
C
=
7
BC = 7
BC
=
7
, and
C
A
=
12
CA = 12
C
A
=
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
A
B
C
ABC
A
BC
?p11. A total of
n
n
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
n
n
?p12. Define
a
∗
b
=
a
+
b
+
a
b
a*b = a + b + ab
a
∗
b
=
a
+
b
+
ab
, for all real numbers
a
,
b
a, b
a
,
b
. If
S
=
{
a
S = \{a
S
=
{
a
real number such that
a
∗
(
−
a
)
>
a
}
a*(-a) > a\}
a
∗
(
−
a
)
>
a
}
, write
S
S
S
as an interval.p13. The diameter of a semicircle coincides with the base
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
. The vertex
C
C
C
moves in such a way that the midpoint of side
A
C
AC
A
C
always lies on the semicircle. What is the curvature of the locus of point
C
C
C
?p14. What is the largest positive integer that divides all the numbers
1
5
−
1
,
2
5
−
2
,
.
.
.
,
n
5
−
n
,
.
.
.
1^5-1, 2^5-2,..., n^5-n,...
1
5
−
1
,
2
5
−
2
,
...
,
n
5
−
n
,
...
?p15. If
2002
=
a
1
+
a
2
⋅
2
!
+
a
3
⋅
3
!
+
.
.
.
+
a
n
⋅
n
!
2002 = a_1 + a_2 \cdot 2! + a_3 \cdot 3! + ...+ a_n \cdot n!
2002
=
a
1
+
a
2
⋅
2
!
+
a
3
⋅
3
!
+
...
+
a
n
⋅
n
!
, where
a
k
a_k
a
k
is an integer,
0
≤
a
k
≤
k
0 \le a_k \le k
0
≤
a
k
≤
k
,
k
=
1
,
2
,
.
.
.
,
n
k = 1, 2, ... , n
k
=
1
,
2
,
...
,
n
, and
a
n
≠
0
a_n \ne 0
a
n
=
0
, find the ordered pair
(
n
,
a
n
)
(n, a_n)
(
n
,
a
n
)
.p16. What is the remainder of the division of
4
3
4
3
43
43^{43^{43}}
4
3
4
3
43
by
100
100
100
?p17. Four couples buy tickets for
8
8
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
8
8
seats?p18. How many
4
4
4
-digit numbers are in the form of
a
b
c
d
‾
\overline{abcd}
ab
c
d
with
a
≤
b
≤
c
≤
d
a \le b \le c \le d
a
≤
b
≤
c
≤
d
?p19. We draw a regular polygon
R
R
R
with
2002
2002
2002
vertices and all its diagonals. How many triangles are formed where all the vertices are vertices of
R
R
R
, but none of the sides are sides of
R
R
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
1
1
st,
2
2
2
nd,
3
3
3
rd,
4
4
4
th,
5
5
5
th,
6
6
6
th finishes obtain consecutive values of
7
,
5
,
4
,
3
,
2
,
1
7, 5, 4, 3, 2, 1
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?
Indonesia Regional MO 2002
Indonesia Regional MO (more commonly known as the provincial level) is the selection for qualifying for the Indonesian MO (National Level) annually. It consists of some structured answers section (short answers) and 5 essay problems. Each short answer problem is worth 1 point, whereas each each essay problem is worth 7 points. The system varies often these days, with the 2021 test (held on 13 September 2021) being 10 essay problems, split into 2 tests.Here I will only be posting the essay problems.Indonesian Regional MO 2002 (Olimpiade Sains Nasional Tingkat Provinsi 2002) Problem 1. Five distinct natural numbers,
k
,
l
,
m
,
n
k, l, m, n
k
,
l
,
m
,
n
and
p
p
p
, will be chosen. The following five information turns out to be sufficient to order the five natural numbers (in ascending, or descending order). (i) Among any two numbers, one of the numbers must divide another, (ii) The integer
m
m
m
is either the largest or the smallest, (iii)
p
p
p
can't divide
m
m
m
and
k
k
k
simultaneously, (iv)
n
≤
l
−
p
n \leq l - p
n
≤
l
−
p
, and (v)
k
k
k
divides
n
n
n
or
p
p
p
divides
n
n
n
, but not both at the same time. Determine all possible orderings of
k
,
l
,
m
,
n
k, l, m, n
k
,
l
,
m
,
n
and
p
p
p
.Problem 2. Determine all positive integers
p
p
p
so that
3
p
+
25
2
p
−
5
\frac{3p+25}{2p-5}
2
p
−
5
3
p
+
25
is also a positive integer.Problem 3. Given a 6-digit number, prove that the six digits can be rearranged in such a way so that the (absolute) difference between the sum of the first and last three digits is no more than 9.Problem 4. It is known the equilateral triangle
A
B
C
ABC
A
BC
and a point
P
P
P
so that the distance between
P
P
P
to
A
A
A
and
C
C
C
is no more than the distance between
P
P
P
to
B
B
B
. Prove that
P
B
=
P
A
+
P
C
PB = PA + PC
PB
=
P
A
+
PC
if and only if
P
P
P
lies on the circumcircle of
△
A
B
C
\triangle{ABC}
△
A
BC
.Problem 5. Consider the tile with the shape of a T-tetromino. Each tile of the tetromino tiles exactly one cell of a checkerboard. We want to cover the checkerboard with tetrominoes so that each tile of the tetromino covers exactly one checkerboard cell, with no overlaps. (a) Prove that we can tile an
8
×
8
8 \times 8
8
×
8
checkerboard, with 16 T-tetrominoes. (b) Prove that we cannot tile a
10
×
10
10 \times 10
10
×
10
checkerboard with 25 T-tetrominoes.(A tetromino is a tile with 4 cells. A T-tetromino is a tetromino with the shape "T".)