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2014 Indonesia Regional
2014 Indonesia Regional
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Indonesia Regional MO 2014 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2013 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685439p23297220]hereTime: 90 minutes
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to be more exact:
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in years 2002-08 time was 90' for part A and 120' for part B
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since years 2009 time is 210' for part A and B totally
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each problem in part A is 1 point, in part B is 7 pointsp1. If
y
=
f
(
x
)
y = f(x)
y
=
f
(
x
)
is a function that satisfies the equation
x
∣
x
∣
+
∣
y
∣
y
=
2
y
\frac{x}{|x|} +\frac{|y|}{y} = 2y
∣
x
∣
x
+
y
∣
y
∣
=
2
y
, then the resulting area of this function is ...p2. If
n
≥
1
n\ge 1
n
≥
1
is a natural number, then the least common multiple of
3
n
−
3
3^n-3
3
n
−
3
and
9
n
+
9
9^n + 9
9
n
+
9
is ...p3. Given a square
A
B
C
D
ABCD
A
BC
D
, point
P
P
P
is inside the square so that
A
P
=
3
AP = 3
A
P
=
3
,
B
P
=
7
BP = 7
BP
=
7
and
D
P
=
5
DP = 5
D
P
=
5
. The area of the square
A
B
C
D
ABCD
A
BC
D
is ...p4. The
n
n
n
-th triangular number is the sum of the first
n
n
n
natural numbers. Define
T
n
T_n
T
n
the sum of the first
n
n
n
triangular numbers. If
T
n
+
x
T
n
+
1
+
y
T
n
−
2
=
n
T_n+xT_{n+1}+yT_{n-2} = n
T
n
+
x
T
n
+
1
+
y
T
n
−
2
=
n
where
x
x
x
and
y
y
y
are integers, then
x
−
y
=
x-y =
x
−
y
=
...p5. Circles
ω
1
\omega_1
ω
1
and
ω
2
\omega_2
ω
2
are tangent at point
A
A
A
and have a common tangent
ℓ
\ell
ℓ
touching
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
in a row at
B
,
C
B,C
B
,
C
. If
B
D
BD
B
D
is diameter of circle
ω
1
\omega_1
ω
1
with length
2
2
2
, and
B
C
=
3
BC = 3
BC
=
3
, the area of triangle
B
D
C
BDC
B
D
C
is ...p6. The last number of 2014 digits of
⌊
6
0
2014
7
⌋
\left\lfloor \frac{60^{2014}}{7} \right\rfloor
⌊
7
6
0
2014
⌋
is...p7. For OSP preparation, a teacher conducts coaching to the students for one week. Every day, during the construction week, each student sends
5
5
5
emails to another student or teacher. At the closing of the show, half of the students got
6
6
6
emails, a third of the students got
4
4
4
emails and the rest one email each. The Master got
2014
2014
2014
email. If the teacher is allowed to take leave during the construction week , then the amount of leave used is ... days. Note: When teachers take leave, students continue to study in the classroom independently and only send emails to fellow students.p8. The sum of all the integers
x
x
x
so that
2
log
(
x
2
−
4
x
−
1
)
^2 \log(x^2-4x-1)
2
lo
g
(
x
2
−
4
x
−
1
)
is integer is...p9. If the roots of the quadratic equation
a
x
2
+
b
x
+
c
=
0
ax^2 + bx + c = 0
a
x
2
+
b
x
+
c
=
0
are in the interval
[
0
,
1
]
[0, 1]
[
0
,
1
]
, then the maximum value of
(
2
a
−
b
)
(
a
−
b
)
a
(
a
−
b
+
c
)
\frac{(2a-b)(a-b)}{a(a- b + c)}
a
(
a
−
b
+
c
)
(
2
a
−
b
)
(
a
−
b
)
is ...p10. All
n
≤
1000
n\le 1000
n
≤
1000
are such that the number
9
+
99
+
999
+
.
.
.
+
99...9
⏟
n
d
i
g
i
t
s
9 + 99 + 999 + ... + \underbrace{99...9}_{n \,\, digits}
9
+
99
+
999
+
...
+
n
d
i
g
i
t
s
99...9
has exactly
n
n
n
digits equal to the number
1
1
1
are ...p11. Given a square
A
B
C
D
ABCD
A
BC
D
with a side length of one unit. Suppose, circle with
A
D
AD
A
D
as diameter, and select point
E
E
E
on side
A
B
AB
A
B
so the line
C
E
CE
CE
is tangent to . The area of triangle
B
C
E
BCE
BCE
is ...p12. A school has four class
11
11
11
study groups. Each study groups send two students to a meeting. They will sit in a circle with no two students from one study group sitting close together. The number of ways is ...(Two ways they sit circular is considered equal if one method can be obtained from the other method another with a rotation).p13. Dono has six cards. Each card is written as a positive integer. For each round, Dono picks
3
3
3
cards at random and adds up the three numbers on the cards. After doing
20
20
20
the possibility of choosing
3
3
3
of
6
6
6
cards, Dono gets a number
16
16
16
10
10
10
times and the number
18
18
18
10
10
10
times. The smallest number that exists on the card is...p14. For real numbers
t
t
t
and positive real numbers
a
a
a
and
b
b
b
satisfy
2
a
2
−
3
a
b
t
+
b
2
=
2
a
2
+
a
b
t
−
b
2
=
0.
2a^2-3abt + b^2 = 2a^2 + abt-b^2 = 0.
2
a
2
−
3
ab
t
+
b
2
=
2
a
2
+
ab
t
−
b
2
=
0.
The value of
t
t
t
is...p15. Let
S
(
n
)
S(n)
S
(
n
)
represent the sum of the digits of
n
n
n
. For example
S
(
567
)
=
5
+
6
+
7
=
18
S(567) = 5 + 6 + 7 = 18
S
(
567
)
=
5
+
6
+
7
=
18
. How many natural numbers
n
n
n
are less than
1000
1000
1000
such that
S
(
n
)
S
(
n
+
1
)
\frac{S(n)}{S(n + 1)}
S
(
n
+
1
)
S
(
n
)
is an integer is ...p16. Given a triangle
A
B
C
ABC
A
BC
, with sides:
A
B
=
c
AB = c
A
B
=
c
,
B
C
=
a
BC = a
BC
=
a
,
C
A
=
b
=
1
2
(
a
+
c
)
CA = b = \frac{1}{2}(a+c)
C
A
=
b
=
2
1
(
a
+
c
)
The largest size of
∠
A
B
C
\angle ABC
∠
A
BC
is...p17. Inside triangle
A
B
C
ABC
A
BC
, draw the point
X
,
Y
,
Z
X, Y,Z
X
,
Y
,
Z
with the rule
∠
X
B
C
=
∠
Z
B
A
=
∠
A
B
C
3
\angle XBC=\angle ZBA = \frac{\angle ABC}{3}
∠
XBC
=
∠
ZB
A
=
3
∠
A
BC
,
∠
X
C
B
=
∠
Y
C
A
=
∠
B
C
A
3
\angle XCB=\angle YCA = \frac{\angle BCA}{3}
∠
XCB
=
∠
Y
C
A
=
3
∠
BC
A
,
∠
Z
A
B
=
∠
Y
A
C
=
∠
B
A
C
3
\angle ZAB=\angle YAC = \frac{\angle BAC}{3}
∠
Z
A
B
=
∠
Y
A
C
=
3
∠
B
A
C
. The length of angle
∠
X
Y
Z
\angle XYZ
∠
X
Y
Z
is...p18. Suppose
0
<
α
,
β
,
γ
<
π
2
0 < \alpha,\beta, \gamma <\frac{\pi}{2}
0
<
α
,
β
,
γ
<
2
π
and
α
+
β
+
γ
=
π
4
\alpha+\beta+ \gamma =\frac{\pi}{4}
α
+
β
+
γ
=
4
π
. The number of triples of positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
so
tan
α
=
1
a
\tan \alpha= \frac{1}{a}
tan
α
=
a
1
,
tan
β
=
1
b
\tan \beta= \frac{1}{b}
tan
β
=
b
1
, and
tan
γ
=
1
c
\tan \gamma = \frac{1}{c}
tan
γ
=
c
1
is...p19. All consecutive odd number triples
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
with
a
<
b
<
c
a < b < c
a
<
b
<
c
such that so that
a
2
+
b
2
+
c
2
a^2 + b^2 + c^2
a
2
+
b
2
+
c
2
is a number with
4
4
4
digits such that are all the digits are the same are ...p20. It is known that a particle in Cartesian coordinates initially lies at the point origin
(
0
,
0
)
(0, 0)
(
0
,
0
)
. The particle is moving, each step is one unit in the same direction positive
X
X
X
-axis, negative
X
X
X
-axis direction, positive
Y
Y
Y
-axis direction, or negative Y-axis direction . The number of ways the particle moves so that after the particle moves 9 steps until the point
(
2
,
3
)
(2, 3)
(
2
,
3
)
is ...
Indonesia Regional MO 2014 Part B
p1. For any number of positive reals
a
,
b
,
c
a, b, c
a
,
b
,
c
with
a
+
b
+
c
=
1
a + b + c = 1
a
+
b
+
c
=
1
, determine the value of
a
b
(
a
2
+
b
2
a
3
+
b
3
)
+
b
c
(
b
2
+
c
2
b
3
+
c
3
)
+
c
a
(
c
2
+
a
2
c
3
+
a
3
)
+
a
4
+
b
4
a
3
+
b
3
+
b
4
+
c
4
b
3
+
c
3
+
c
4
+
a
4
c
3
+
a
3
ab\left(\frac{a^2+b^2}{a^3+b^3}\right)+bc\left(\frac{b^2+c^2}{b^3+c^3}\right)+ca\left(\frac{c^2+a^2}{c^3+a^3}\right)+\frac{a^4+b^4}{a^3+b^3}+\frac{b^4+c^4}{b^3+c^3}+\frac{c^4+a^4}{c^3+a^3}
ab
(
a
3
+
b
3
a
2
+
b
2
)
+
b
c
(
b
3
+
c
3
b
2
+
c
2
)
+
c
a
(
c
3
+
a
3
c
2
+
a
2
)
+
a
3
+
b
3
a
4
+
b
4
+
b
3
+
c
3
b
4
+
c
4
+
c
3
+
a
3
c
4
+
a
4
[url=https://artofproblemsolving.com/community/c6h2371565p19388497]p2. Given an acute triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB <AC
A
B
<
A
C
. The ex-circles of triangle
A
B
C
ABC
A
BC
opposite
B
B
B
and
C
C
C
are centered on
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively. Let
D
D
D
be the midpoint of
B
1
C
1
B_1C_1
B
1
C
1
. Suppose that
E
E
E
is the point of intersection of
A
B
AB
A
B
and
C
D
CD
C
D
, and
F
F
F
is the point of intersection of
A
C
AC
A
C
and
B
D
BD
B
D
. If
E
F
EF
EF
intersects
B
C
BC
BC
at point
G
G
G
, prove that
A
G
AG
A
G
is the bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
. p3. It is known that
X
X
X
is a set with
102
102
102
elements. Suppose
A
1
,
A
2
,
.
.
.
,
A
101
A_1, A_2, ..., A_{101}
A
1
,
A
2
,
...
,
A
101
is a set of subset of
X
X
X
such that the sum of every
50
50
50
of them has more than
100
100
100
elements. Prove that there are
1
≤
i
<
j
<
k
≤
101
1 \le i <j <k \le 101
1
≤
i
<
j
<
k
≤
101
such that
A
i
∩
A
j
A_i \cap A_j
A
i
∩
A
j
,
A
i
∩
A
k
A_i \cap A_k
A
i
∩
A
k
and
A
k
∩
A
j
A_k \cap A_j
A
k
∩
A
j
are not empty.[url=https://artofproblemsolving.com/community/c6h2371573p19388630]p4. Let
Γ
\Gamma
Γ
be the circumcircle of triangle
A
B
C
ABC
A
BC
. One circle
ω
\omega
ω
is tangent to
Γ
\Gamma
Γ
at
A
A
A
and tangent to
B
C
BC
BC
at
N
N
N
. Suppose that the extension of
A
N
AN
A
N
crosses
Γ
\Gamma
Γ
again at
E
E
E
. Let
A
D
AD
A
D
and
A
F
AF
A
F
be respectively the line of altitude
A
B
C
ABC
A
BC
and diameter of
Γ
\Gamma
Γ
, show that
A
N
×
A
E
=
A
D
×
A
F
=
A
B
×
A
C
AN \times AE = AD \times AF = AB \times AC
A
N
×
A
E
=
A
D
×
A
F
=
A
B
×
A
C
p5. Suppose
{
a
n
}
\{a_n\}
{
a
n
}
is a sequence of integers that satisfies
a
1
=
2
a_1 = 2
a
1
=
2
,
a
2
=
8
a_2 = 8
a
2
=
8
and
a
n
+
2
=
3
a
n
+
1
−
a
n
+
5
(
−
1
)
n
.
a_{n+2} = 3a_{n+1}-a_n + 5(-1)^n.
a
n
+
2
=
3
a
n
+
1
−
a
n
+
5
(
−
1
)
n
.
a) Is
a
2014
a_{2014}
a
2014
prime? b) Prove that for every odd number
m
m
m
, the number
a
m
+
a
4
m
a
2
m
+
a
3
m
\frac{a_m + a_{4m}}{a_{2m} + a_{3m}}
a
2
m
+
a
3
m
a
m
+
a
4
m
is an integer.