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Indonesia Regional MO 2006 Part A 20 problems 90' , answer only
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2006 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2684669p23288980]hereTime: 90 minutes
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Each question is worth 1 (one) point. p1. The sum of all integers between
2006
3
\sqrt[3]{2006}
3
2006
and
2006
\sqrt{2006}
2006
is ...p2. In trapezoid
A
B
C
D
ABCD
A
BC
D
, side
A
B
AB
A
B
is parallel to
D
C
DC
D
C
. A circle tangent to all four sides of the trapezoid can be drawn. If
A
B
=
75
AB = 75
A
B
=
75
and
D
C
=
40
DC = 40
D
C
=
40
, then the perimeter of the trapezoid
A
B
C
D
=
.
.
.
ABCD = ...
A
BC
D
=
...
p3. The set of all
x
x
x
that satisfies
(
x
−
1
)
3
+
(
x
−
2
)
2
=
1
(x-1)^3 + (x-2)^2 = 1
(
x
−
1
)
3
+
(
x
−
2
)
2
=
1
is ...p4. The largest two-digit prime number which is the sum of two other prime numbers is ...p5. Afkar selects the terms of an infinite geometric sequence
1
,
1
2
,
1
4
,
1
8
1, \frac12, \frac14, \frac18
1
,
2
1
,
4
1
,
8
1
, to create a new infinite geometric sequence whose sum is
1
7
\frac17
7
1
. The first three terms Afkar chooses are ...p6. The area of the sides of a cuboid are
486
486
486
,
486
486
486
,
243
243
243
,
243
243
243
,
162
162
162
,
162
162
162
. The volume of the cuboid is ...p7. The maximum value of the function
f
(
x
)
=
(
1
3
)
x
2
−
4
x
+
3
f(x) = \left(\frac13 \right)^{x^2-4x+3}
f
(
x
)
=
(
3
1
)
x
2
−
4
x
+
3
is ...p8. Given the function
f
(
x
)
=
∣
∣
x
−
2
∣
−
a
∣
−
3
f(x) = ||x - 2| - a| -3
f
(
x
)
=
∣∣
x
−
2∣
−
a
∣
−
3
. If the graph f intersects the
x
x
x
-axis at exactly three points, then
a
=
.
.
.
a =. ..
a
=
...
p9. For natural numbers
n
n
n
, write
s
(
n
)
=
1
+
2
+
.
.
.
+
n
s(n) = 1 + 2 + ...+ n
s
(
n
)
=
1
+
2
+
...
+
n
and
p
(
n
)
=
1
×
2
×
.
.
.
×
n
p(n) = 1 \times 2 \times ... \times n
p
(
n
)
=
1
×
2
×
...
×
n
. The smallest even number
n
n
n
such that
p
(
n
)
p(n)
p
(
n
)
is divisible by
s
(
n
)
s(n)
s
(
n
)
is ...p10. If
∣
x
∣
+
x
+
y
=
10
|x|+ x + y = 10
∣
x
∣
+
x
+
y
=
10
and
x
+
∣
y
∣
−
y
=
12
x + |y|-y = 12
x
+
∣
y
∣
−
y
=
12
, then
x
+
y
=
.
.
.
x + y = ...
x
+
y
=
...
p11. A set of three natural numbers is called an arithmetic set if one of its elements is the average of the other two elements. The number of arithmetic subsets of
{
1
,
2
,
3
,
.
.
.
,
8
}
\{1,2,3,...,8\}
{
1
,
2
,
3
,
...
,
8
}
is ...p12. From each one-digit number
a
a
a
, the number
N
N
N
is made by juxtaposing the three numbers
a
+
2
a + 2
a
+
2
,
a
+
1
a + 1
a
+
1
,
a
a
a
i.e.
N
=
(
a
+
2
)
(
a
+
1
)
a
‾
N = \overline{(a+2)(a+1)a}
N
=
(
a
+
2
)
(
a
+
1
)
a
. For example, for
a
=
8
a = 8
a
=
8
,
N
=
1098
N = 1098
N
=
1098
. The ten such numbers
N
N
N
have the greatest common divisor ...p13. If
x
2
+
1
x
2
=
47
x^2+\frac{1}{x^2}=47
x
2
+
x
2
1
=
47
, then
x
+
1
x
=
.
.
.
\sqrt{x}+\frac{1}{\sqrt{x}}= ...
x
+
x
1
=
...
p14. A class will choose a student from among them to represent the class. Every student has the same opportunity to be selected. The probability that a male student is selected is
2
3
\frac23
3
2
times the probability that a female student is selected. The percentage of male students in the class is ...p15. In triangle
A
B
C
ABC
A
BC
, the bisector of angle
A
A
A
intersects side
B
C
BC
BC
at point D. If
A
B
=
A
D
=
2
AB = AD = 2
A
B
=
A
D
=
2
and
B
D
=
1
BD = 1
B
D
=
1
, then
C
D
=
.
.
.
CD = ...
C
D
=
...
p16. If
(
x
−
1
)
2
(x-1)^2
(
x
−
1
)
2
divides
a
x
4
+
b
x
3
+
1
ax^4 + bx^3 + 1
a
x
4
+
b
x
3
+
1
, then
a
b
=
.
.
.
ab = ...
ab
=
...
p17. From point
O
O
O
, two half-lines (rays)
l
1
l_1
l
1
and
l
2
l_2
l
2
are drawn which form an acute angle . The different points
A
1
,
A
3
,
A
5
A_1, A_3, A_5
A
1
,
A
3
,
A
5
lie on the line
l
2
l_2
l
2
, while the points
A
2
,
A
4
,
A
6
A_2, A_4, A_6
A
2
,
A
4
,
A
6
lie on the line
l
1
l_1
l
1
. If
A
1
A
2
=
A
2
A
3
=
A
3
A
4
=
A
4
O
=
O
A
5
=
A
5
A
6
=
A
6
A
1
A_1A_2 = A_2A_3 = A_3A_4 = A_4O = OA_5 = A_5A_6 = A_6A_1
A
1
A
2
=
A
2
A
3
=
A
3
A
4
=
A
4
O
=
O
A
5
=
A
5
A
6
=
A
6
A
1
, then = ...p18. The number of different
7
7
7
-digit numbers that can be formed by changing the order of the numbers
2504224
2504224
2504224
is ...p19. Evan creates a sequence of natural numbers
a
1
,
a
2
,
a
3
a_1, a_2, a_3
a
1
,
a
2
,
a
3
, which satisfies
a
k
+
1
−
a
k
=
2
(
a
k
−
a
k
−
1
)
−
1
a_{k+1}-a_k = 2(a_k-a_{k-1})-1
a
k
+
1
−
a
k
=
2
(
a
k
−
a
k
−
1
)
−
1
, for
k
=
2
,
3
,
.
.
.
,
k = 2, 3,...,
k
=
2
,
3
,
...
,
and
a
2
−
a
1
=
2
a_2 - a_1 = 2
a
2
−
a
1
=
2
. If
2006
2006
2006
appears in the sequence, the smallest possible value of
a
1
a_1
a
1
is ...p20. In triangle
A
B
C
ABC
A
BC
, the medians from vertex
B
B
B
and vertex
C
C
C
intersect at right angles to each other. The minimum value of
cot
B
+
cot
C
\cot B + \cot C
cot
B
+
cot
C
is ...
Indonesia Regional MO 2006 Part B
[url=https://artofproblemsolving.com/community/c6h2372254p19397375]p1. Suppose that triangle
A
B
C
ABC
A
BC
is a right angled triangle at
B
B
B
. The altitude from
B
B
B
intersects the side
A
C
AC
A
C
at point
D
D
D
. If the points
E
E
E
and
F
F
F
are the midpoints of
B
D
BD
B
D
and
C
D
CD
C
D
, respectively, prove that
A
E
⊥
B
F
AE \perp BF
A
E
⊥
BF
.p2. Let
m
m
m
be a natural number that satisfy
1003
<
m
<
2006
1003 < m < 2006
1003
<
m
<
2006
. Given the set of natural numbers
S
=
{
1
,
2
,
3
,
.
.
.
,
m
}
S = \{1, 2, 3,..., m\}
S
=
{
1
,
2
,
3
,
...
,
m
}
, how many members of
S
S
S
must be selected so that there is always at least one pair of selected elements has sum
2006
2006
2006
?p3. Let
d
=
g
c
d
(
7
n
+
5
,
5
n
+
4
)
d = gcd (7n + 5, 5n + 4)
d
=
g
c
d
(
7
n
+
5
,
5
n
+
4
)
, where n is a natural number. (a) Prove that for every natural number
n
n
n
,
d
=
1
d = 1
d
=
1
or
3
3
3
. (b) Prove that
d
=
3
d = 3
d
=
3
if and only if
n
=
3
k
+
1
n = 3k + 1
n
=
3
k
+
1
, for a natural number
k
k
k
.p4. Win has two coins. He will do the following procedure over and over again as long as he still has coins: toss all the coins he has at the same time; every coin that appears with a number side will be given to Albert. Determine the probability that Win will repeat this procedure more than three times.p5. Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be natural numbers. If all the three roots of the equation
x
2
−
2
a
x
+
b
=
0
x^2 - 2ax + b = 0
x
2
−
2
a
x
+
b
=
0
x
2
−
2
b
x
+
c
=
0
x^2 -2bx + c = 0
x
2
−
2
b
x
+
c
=
0
x
2
−
2
c
x
+
a
=
0
x^2 - 2cx + a = 0
x
2
−
2
c
x
+
a
=
0
are natural numbers, determine
a
,
b
a, b
a
,
b
and
c
c
c
.