MathDB
Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia Regional
2008 Indonesia Regional
2008 Indonesia Regional
Part of
Indonesia Regional
Subcontests
(1)
2
Hide problems
Indonesia Regional MO 2008 Part B
p1. Find all pairs of natural numbers
(
x
,
n
)
(x, n)
(
x
,
n
)
that satisfy
1
+
x
+
x
2
+
.
.
.
+
x
n
=
40
1 + x + x^2 + ... + x^n = 40
1
+
x
+
x
2
+
...
+
x
n
=
40
p2. Given a real polynomial
P
(
x
)
=
x
2008
+
a
1
x
2007
+
a
2
x
2006
+
.
.
.
+
a
2007
x
+
a
2008
P(x) = x^{2008} + a_1x^{2007} + a_2x^{2006} + ...+ a_{2007}x + a_{2008}
P
(
x
)
=
x
2008
+
a
1
x
2007
+
a
2
x
2006
+
...
+
a
2007
x
+
a
2008
and
Q
(
x
)
=
x
2
+
2
x
+
2008
Q(x) = x^2 + 2x + 2008
Q
(
x
)
=
x
2
+
2
x
+
2008
. Suppose the equation
P
(
x
)
=
0
P(x) = 0
P
(
x
)
=
0
has
2008
2008
2008
real solutions and
P
(
2008
)
≤
1
P(2008)\le 1
P
(
2008
)
≤
1
. Show that the equation
P
(
Q
(
x
)
)
=
0
P(Q(x)) = 0
P
(
Q
(
x
))
=
0
has a real solution.[url=https://artofproblemsolving.com/community/c6h211746p1167343]p3. The inscribed circle of triangle ABC, touches the sides
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
at
D
D
D
,
E
E
E
, and
F
F
F
, respectively. Through
D
D
D
, a perpendicular line
E
F
EF
EF
is drawn that intersects
E
F
EF
EF
at
G
G
G
. Prove that \frac{FG}{EG}=\frac{BF}{CE}.p4. The numbers
1
,
2
,
3
,
.
.
.
,
9
1, 2, 3, ..., 9
1
,
2
,
3
,
...
,
9
are arranged in a circle randomly. Prove that there are three adjacent numbers whose sum is greater than
15
15
15
.5. Determine the number of positive
5
5
5
-digit palindromes that are divisible by
3
3
3
. A palindrome is a number/word that is the same if read from left to right or vice versa. For example,
35353
35353
35353
is a palindrome, while
14242
14242
14242
is not.
Indonesia Regional MO 2008 Part A 20 problems 90' , answer only
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2008 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684687p23289130]hereTime: 90 minutes
∙
\bullet
∙
Write only the answers to the questions given.
∙
\bullet
∙
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.
∙
\bullet
∙
Each question is worth 1 (one) point. p1. The number of positive divisors from
2008
2008
2008
is ...p2. How many ways are there to arrange letters of word MATEMATIKA with the two T's not adjacent? ...p3. If
0
<
b
<
a
0 < b < a
0
<
b
<
a
and
a
2
+
b
2
=
6
a
b
a^2 + b^2 = 6ab
a
2
+
b
2
=
6
ab
, then
a
+
b
a
−
b
=
.
.
.
\frac{a+b}{a-b}= ...
a
−
b
a
+
b
=
...
p4. Two of the altitudes of triangle
A
B
C
ABC
A
BC
are acute, equal to
4
4
4
and
12
12
12
, respectively. If the length of the third altitude of the triangle is an integer, then the maximum length of the height of the triangle is ...p5. In the
X
O
Y
XOY
XO
Y
plane, the number of lines that intersect the
X
X
X
axis at the point with the abscissa of prime numbers and intersect the
Y
Y
Y
axis at the point with positive integer ordinates and through the point
(
4
,
3
)
(4, 3)
(
4
,
3
)
is ...p6. Given a triangle
A
B
C
ABC
A
BC
,
A
D
AD
A
D
is perpendicular to
B
C
BC
BC
such that
D
C
=
2
DC = 2
D
C
=
2
and
B
D
=
3
BD = 3
B
D
=
3
. If
∠
B
A
C
=
4
5
o
\angle BAC = 45^o
∠
B
A
C
=
4
5
o
, then the area of triangle
A
B
C
ABC
A
BC
is ...p7. If
x
x
x
and
y
y
y
are integers that satisfy
y
2
+
3
x
2
y
2
=
30
x
2
+
517
y^2 + 3x^2y^2 = 30x^2 + 517
y
2
+
3
x
2
y
2
=
30
x
2
+
517
, then
3
x
2
y
2
=
.
.
.
3x^2y^2 = ...
3
x
2
y
2
=
...
p8. Given a triangle
A
B
C
ABC
A
BC
, where
B
C
=
a
BC = a
BC
=
a
,
A
C
=
b
AC = b
A
C
=
b
and
∠
C
=
6
0
o
\angle C = 60^o
∠
C
=
6
0
o
. If
a
b
=
2
+
3
\frac{a}{b}=2+\sqrt3
b
a
=
2
+
3
, then the measure of angle
B
B
B
is ...p9. One hundred students from a province in Java took part in the selection at the provincial level and the average score was
100
100
100
. The number of grade II students who took part in the selection was
50
%
50\%
50%
more than grade III students, and the average score of grade III students was
50
%
50\%
50%
higher of the average score of class II students. The average score of class III students is ...p10. Given triangle
A
B
C
ABC
A
BC
, where
B
C
=
5
BC = 5
BC
=
5
,
A
C
=
12
AC = 12
A
C
=
12
, and
A
B
=
13
AB = 13
A
B
=
13
. Points
D
D
D
and
E
E
E
on
A
B
AB
A
B
and
A
C
AC
A
C
respectivrly are such that
D
E
DE
D
E
divides triangle
A
B
C
ABC
A
BC
into two equal parts. The minimum length of
D
E
DE
D
E
is ...p11. Let
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
be rational numbers. If it is known that the equation
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
=
0
x^4 + ax^3 + bx^2 + cx + d = 0
x
4
+
a
x
3
+
b
x
2
+
c
x
+
d
=
0
has
4
4
4
real roots, two of which are
2
\sqrt2
2
and
2008
\sqrt{2008}
2008
. The value of
a
+
b
+
c
+
d
a + b + c + d
a
+
b
+
c
+
d
is ...p12. Given a triangle
A
B
C
ABC
A
BC
with sides
a
,
b
a, b
a
,
b
, and
c
c
c
. The value of
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
is equal to
16
16
16
times the area of triangle
A
B
C
ABC
A
BC
. The value of
cot
A
+
cot
B
+
cot
C
\cot A + \cot B + \cot C
cot
A
+
cot
B
+
cot
C
is ...p13. Given
f
(
x
)
=
x
2
+
4
f(x) = x^2 + 4
f
(
x
)
=
x
2
+
4
. Let
x
x
x
and
y
y
y
be positive real numbers that satisfy
f
(
x
y
)
+
f
(
y
−
x
)
=
f
(
y
+
x
)
f(xy) + f(y-x) = f(y+x)
f
(
x
y
)
+
f
(
y
−
x
)
=
f
(
y
+
x
)
. The minimum value of
x
+
y
x + y
x
+
y
is ...p14. The number of positive integers
n
n
n
less than
2008
2008
2008
that has exactly
n
2
\frac{n}{2}
2
n
numbers less than
n
n
n
and is prime relative to
n
n
n
is ...p15. A polynomial
f
(
x
)
f(x)
f
(
x
)
satisfies the equation
f
(
x
2
)
−
x
3
f
(
x
)
=
2
(
x
3
−
1
)
f(x^2)-x^3f(x) = 2(x^3-1)
f
(
x
2
)
−
x
3
f
(
x
)
=
2
(
x
3
−
1
)
for every
x
x
x
real number. The degree (highest power of
x
x
x
) of
f
(
x
)
f(x)
f
(
x
)
is ..p16. Assume one year
365
365
365
days. The probability that out of
20
20
20
people chosen at random, two people have a birthday on the same day is ...p17. Three numbers are chosen at random from
{
1
,
2
,
3
,
.
.
.
,
2008
}
\{1,2,3,...,2008\}
{
1
,
2
,
3
,
...
,
2008
}
. The probability that the sum of all three is even is ...p18. Let
∣
X
∣
|X|
∣
X
∣
represent the number of members of the set
X
X
X
. If
∣
A
∪
B
∣
=
10
|A\cup B| = 10
∣
A
∪
B
∣
=
10
and
∣
A
∣
=
4
|A| = 4
∣
A
∣
=
4
, then the possible values for
B
B
B
are ...p19. It is known that
A
D
AD
A
D
is the altitude of triangle
A
B
C
ABC
A
BC
,
∠
D
A
B
=
∠
A
C
D
\angle DAB = \angle ACD
∠
D
A
B
=
∠
A
C
D
,
A
D
=
6
AD = 6
A
D
=
6
,
B
D
=
8
BD = 8
B
D
=
8
. The area of triangle
A
B
C
ABC
A
BC
is ...p20. Find the value of
∑
k
=
0
1004
3
k
(
1004
k
)
\sum_{k=0}^{1004} 3^k {{1004} \choose k}
∑
k
=
0
1004
3
k
(
k
1004
)
.