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2007 Indonesia Regional
2007 Indonesia Regional
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Indonesia Regional MO 2007 Part A 20 problems 90' , answer only
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]hereTime: 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
4
4
-digit odd number who is sum of all prime numbers is ...p2. The amount of money consists of coins in the
500
500
500
Rp.,
200
200
200
Rp. , and
100
100
100
Rp. with a total value of
100
,
000
100,000
100
,
000
Rp.. If the
500
500
500
's note is half the
200
200
200
's note, but three times the
100
100
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
1
1
unit longer than the length of the shortest side. The area of the triangle is ... square units .p4. Among the numbers
2006
2006
2006
,
2007
2007
2007
and
2008
2008
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
%
10\%
10%
on the first car, but breaks even (return on capital) for both cars. The merchant's profit percentage for the second car isp6. 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
245
245
cm
2
^2
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
3
3
points if it wins,
0
0
0
if it loses and
1
1
1
if the match ends in a draw. At the end of the tournament one of the teams gets a total score of
4
4
4
. The sum of the total points of the other three teams is at least ...p8. For natural numbers
n
n
n
. In its simplest form,
1
!
1
+
2
!
2
+
3
!
3
+
.
.
.
+
n
!
n
=
.
.
.
1!1+2!2+3!3+...+n!n=...
1
!
1
+
2
!
2
+
3
!
3
+
...
+
n
!
n
=
...
p9. Point
P
P
P
is located in quadrant I on the line
y
=
x
y = x
y
=
x
. The point
Q
Q
Q
lies on the line
y
=
2
x
y = 2x
y
=
2
x
such that
P
Q
PQ
PQ
is perpendicular to the line
y
=
x
y = x
y
=
x
and
P
Q
=
2
PQ = 2
PQ
=
2
. Then the coordinates of
Q
Q
Q
are ...p10. The set of all natural numbers
n
n
n
such that
6
n
+
30
6n + 30
6
n
+
30
is a multiple of
2
n
+
1
2n + 1
2
n
+
1
is ...p11. The term constant in the
(
2
x
2
−
1
x
)
9
\left(2x^2-\frac{1}{x}\right)^9
(
2
x
2
−
x
1
)
9
expansion is ...p12. The abscissa of the point where the line
ℓ
\ell
ℓ
intersects the
x
x
x
-axis and the coordinates of the point where it intersects
ℓ
\ell
ℓ
with the
y
y
y
-axis are prime numbers. If
ℓ
\ell
ℓ
also passes through the point
(
3
,
4
)
(3, 4)
(
3
,
4
)
, the equation of
ℓ
\ell
ℓ
is ...p13. Seventeen candies are packed into bags so that the number of candies in each two bags is at most
1
1
1
. The number of ways to pack the candies into at least two bags is ...p14. If the maximum value of
x
+
y
x + y
x
+
y
in the set
{
(
x
,
y
)
∣
x
≥
0
,
y
≥
0
,
x
+
3
y
≤
6
,
3
x
+
y
≤
a
}
\{(x, y)|x\ge 0, y\ge0, x + 3y \le 6, 3x + y \le a\}
{(
x
,
y
)
∣
x
≥
0
,
y
≥
0
,
x
+
3
y
≤
6
,
3
x
+
y
≤
a
}
is
4
4
4
, then
a
=
.
.
.
a = ...
a
=
...
p15. A
5
×
5
×
5
5 \times 5 \times 5
5
×
5
×
5
cube is composed of
125
125
125
unit cubes. The surface of the large cube is then painted. The ratio of the sides (surfaces) of the
125
125
125
painted to unpainted unit cubes is ....p16. A square board is divided into
4
×
4
4 \times 4
4
×
4
squares and colored like a chessboard. Each tile is numbered from
1
1
1
to
16
16
16
. Andi wants to cover the tiles on the board with
7
7
7
cards of
2
×
1
2 \times 1
2
×
1
tile. In order for his
7
7
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
1, 2, ... , n
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
7
17
35 \frac{7}{17}
35
17
7
. The
n
n
n
-th possible number for this to happen is ...p18. Given a triangle
A
B
C
ABC
A
BC
right at
A
A
A
, point
D
D
D
at
A
C
AC
A
C
and point
F
F
F
at
B
C
BC
BC
. If
A
F
⊥
B
C
AF\perp BC
A
F
⊥
BC
and
B
D
=
D
C
=
F
C
=
1
BD = DC = FC = 1
B
D
=
D
C
=
FC
=
1
, then
A
C
=
.
.
.
AC = ...
A
C
=
...
p19. Among all the natural number solutions
(
x
,
y
)
(x, y)
(
x
,
y
)
of the equation
x
+
y
2
+
x
y
=
54
\frac{x+y}{2}+\sqrt{xy}=54
2
x
+
y
+
x
y
=
54
, the solution with the largest
x
x
x
is
(
x
,
y
)
=
.
.
.
(x, y) = ...
(
x
,
y
)
=
...
p20. Let
V
V
V
be the set of points on the plane with integer coordinates and
X
X
X
be the set of midpoints of all pairs of points in the set
V
V
V
. To ensure that any member of
X
X
X
also has integer coordinates, the number of members of
V
V
V
must be at least ...
Indonesia Regional MO 2007 Part B
[url=https://artofproblemsolving.com/community/c4h2372294p19397629]p1. Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral with
A
B
=
B
C
=
C
D
=
D
A
AB = BC = CD = DA
A
B
=
BC
=
C
D
=
D
A
. (a) Prove that point
A
A
A
must be outside the triangle
B
C
D
BCD
BC
D
. (b) Prove that every pair of opposite sides of
A
B
C
D
ABCD
A
BC
D
is always parallel.p2. Suppose
a
a
a
and
b
b
b
are two natural numbers, one of which is not a multiple of the other. Let's also say that
l
c
m
(
a
,
b
)
lcm(a, b)
l
c
m
(
a
,
b
)
is a
2
2
2
-digit number, while
g
c
d
(
a
,
b
)
gcd(a, b)
g
c
d
(
a
,
b
)
can be obtained by reversing the order of numbers in
l
c
m
(
a
,
b
)
lcm(a, b)
l
c
m
(
a
,
b
)
. Find the largest possible
b
b
b
.p3. Find all real numbers
x
x
x
that satisfy
x
4
−
4
x
3
+
5
x
2
−
4
x
+
1
=
0
x^4 -4x^3 + 5x^2 - 4x + 1 = 0
x
4
−
4
x
3
+
5
x
2
−
4
x
+
1
=
0
[url=https://artofproblemsolving.com/community/c6h2372255p19397376]p4. In acute triangle
A
B
C
ABC
A
BC
,
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
are the altitudes, with D, E, F on sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
, respectively. Prove that
D
E
+
D
F
≤
B
C
DE + DF \le BC
D
E
+
D
F
≤
BC
p5. The numbers
1
,
2
,
3
,
.
.
.
,
15
,
16
1, 2, 3,..., 15, 16
1
,
2
,
3
,
...
,
15
,
16
are arranged on a
4
×
4
4 \times 4
4
×
4
. For
i
=
1
,
2
,
3
,
4
i = 1, 2, 3, 4
i
=
1
,
2
,
3
,
4
, let
b
i
b_i
b
i
be the sum of the numbers in the
i
i
i
-th row and
k
i
k_i
k
i
is the sum of the numbers in column
i
i
i
. Also, let
d
1
d_1
d
1
and
d
2
d_2
d
2
be the sum of the numbers on the two diagonals. The arrangement can be called antimagic if
b
1
b_1
b
1
,
b
2
b_2
b
2
,
b
3
b_3
b
3
,
b
4
b_4
b
4
,
k
1
k_1
k
1
,
k
2
k_2
k
2
,
k
3
k_3
k
3
,
k
4
k_4
k
4
,
d
1
d_1
d
1
,
d
2
d_2
d
2
can be arranged into ten consecutive numbers. Determine the largest number among these ten consecutive numbers that can be obtained from an antimagic.