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2015 Indonesia Regional
2015 Indonesia Regional
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Indonesia Regional MO 2015 Part B
p1. Let
X
=
{
1
,
2
,
3
,
4
,
5
}
X = \{1, 2, 3, 4, 5\}
X
=
{
1
,
2
,
3
,
4
,
5
}
. Let
F
=
{
A
1
,
A
2
,
A
3
,
.
.
.
,
A
m
}
F = \{A_1,A_2,A_3, ..., A_m\}
F
=
{
A
1
,
A
2
,
A
3
,
...
,
A
m
}
, with
A
i
⊆
X
A_i\subseteq X
A
i
⊆
X
and
2
2
2
members in
A
i
A_i
A
i
, for
i
=
1
,
2
,
.
.
.
,
m
i = 1,2,...,m
i
=
1
,
2
,
...
,
m
. Determine the minimum
m
m
m
so that for any
B
⊆
X
B\subseteq X
B
⊆
X
, with
B
B
B
having
3
3
3
members, there is a member of
F
F
F
contained in
B
B
B
. Prove your answer.p2. Find all triples of real numbers
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
that satisfy the system of equations:
(
x
+
1
)
2
=
x
+
y
+
2
(x + 1)^2 = x + y + 2
(
x
+
1
)
2
=
x
+
y
+
2
(
y
+
1
)
2
=
y
+
z
+
2
(y + 1)^2 = y + z + 2
(
y
+
1
)
2
=
y
+
z
+
2
(
z
+
1
)
2
=
z
+
x
+
2
(z + 1)^2 = z + x + 2
(
z
+
1
)
2
=
z
+
x
+
2
[url=https://artofproblemsolving.com/community/c6h2371594p19389035]p3. Given the isosceles triangle
A
B
C
ABC
A
BC
, where
A
B
=
A
C
AB = AC
A
B
=
A
C
. Let
D
D
D
be a point in the segment
B
C
BC
BC
so that
B
D
=
2
D
C
BD = 2DC
B
D
=
2
D
C
. Suppose also that point
P
P
P
lies on the segment
A
D
AD
A
D
such that:
∠
B
A
C
=
∠
B
P
D
\angle BAC = \angle BP D
∠
B
A
C
=
∠
BP
D
. Prove that
∠
B
A
C
=
2
∠
D
P
C
\angle BAC = 2\angle DP C
∠
B
A
C
=
2∠
D
PC
. p4. Suppose
p
1
,
p
2
,
.
.
,
p
n
p_1, p_2,.., p_n
p
1
,
p
2
,
..
,
p
n
arithmetic sequence with difference
b
>
0
b > 0
b
>
0
and
p
i
p_i
p
i
prime for each
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2, ..., n
i
=
1
,
2
,
...
,
n
. a) If
p
1
>
n
p_1 > n
p
1
>
n
, prove that every prime number
p
p
p
with
p
≤
n
p\le n
p
≤
n
, then
p
p
p
divides
b
b
b
. b) Give an example of an arithmetic sequence
p
1
,
p
2
,
.
.
,
p
10
p_1, p_2,.., p_{10}
p
1
,
p
2
,
..
,
p
10
, with positive difference and
p
i
p_i
p
i
prime for
i
=
1
,
2
,
.
.
.
,
10
i = 1, 2, ..., 10
i
=
1
,
2
,
...
,
10
. p5. Given a set consisting of
22
22
22
integers,
A
=
{
±
a
1
,
±
a
2
,
.
.
.
,
±
a
11
}
A = \{\pm a_1, \pm a_2, ..., \pm a_{11}\}
A
=
{
±
a
1
,
±
a
2
,
...
,
±
a
11
}
. Prove that there is a subset
S
S
S
of
A
A
A
which simultaneously has the following properties: a) For each
i
=
1
,
2
,
.
.
.
,
11
i = 1, 2, ..., 11
i
=
1
,
2
,
...
,
11
at most only one of the
a
i
a_i
a
i
or
−
a
i
-a_i
−
a
i
is a member of
S
S
S
b) The sum of all the numbers in
S
S
S
is divided by
2015
2015
2015
.
Indonesia Regional MO 2015 Part A
Indonesia Regional also know as provincial level, is a qualifying round for National Math Olympiad Year 2015 [hide=Part A]Part B consists of 5 essay / proof problems, posted [url=https://artofproblemsolving.com/community/c4h2685462p23297434]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.
∙
∙
\bullet \bullet
∙
∙
to be more exact:
⊳
\rhd
⊳
in years 2002-08 time was 90' for part A and 120' for part B
⊳
\rhd
⊳
since years 2009 time is 210' for part A and B totally
⊳
\rhd
⊳
each problem in part A is 1 point, in part B is 7 points p1. The sum of all the real numbers x that satisfy
x
2
−
2
x
=
2
+
x
x
2
−
4
x
x^2-2x = 2 + x\sqrt{x^2-4x}
x
2
−
2
x
=
2
+
x
x
2
−
4
x
is ... p2. The number of integers
n
n
n
, so that
n
+
1
n + 1
n
+
1
is a factor of
n
2
+
1
n^2 + 1
n
2
+
1
is ... p3. At a party, each man shakes hands with another man only once. Likewise, each woman only shakes hands once with another woman who attended the party. No shaking hands between men and the women at the party. If many men are present at the party more than women and the number of handshakes between men or women is there
7
7
7
handshakes. The number of men present at the party is... p4. Given a triangle
A
B
C
ABC
A
BC
, through the point
D
D
D
which lies on the side
B
C
BC
BC
are drawn the lines
D
E
DE
D
E
and
D
F
DF
D
F
, parallel to
A
B
AB
A
B
and
A
C
AC
A
C
, respectively, (
E
E
E
in
A
C
AC
A
C
,
F
F
F
on
A
B
AB
A
B
). If the area of triangle
D
E
C
DEC
D
EC
is
4
4
4
times the area of triangle
B
D
F
BDF
B
D
F
, then the ratio of the area of triangle
A
E
F
AEF
A
EF
to the area of triangle
A
B
C
ABC
A
BC
is ... p5. If
f
f
f
is a function defined on the set of real numbers, such that
3
f
(
x
)
−
2
f
(
2
−
x
)
=
x
2
+
8
x
−
9
3f(x)-2f(2-x) = x^2 + 8x-9
3
f
(
x
)
−
2
f
(
2
−
x
)
=
x
2
+
8
x
−
9
for all real numbers
x
x
x
, then the value of
f
(
2015
)
f(2015)
f
(
2015
)
is ... p6. The number of pairs of integers
(
a
,
b
)
(a, b)
(
a
,
b
)
that satisfy
1
a
+
1
b
+
1
=
1
2015
\frac{1}{a}+\frac{1}{b + 1}=\frac{1}{2015}
a
1
+
b
+
1
1
=
2015
1
is ... p7. There are
10
10
10
people, five boys and five girls, including a couple of bride. A photographer who is not one of the
10
10
10
people will take pictures of six of them, including the two brides, with neither two men nor two women close together. The number of ways is ... p8. The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The value of the cosine of the smallest angle is ... p9. Given two different squared terms
f
(
x
)
=
x
2
+
a
x
+
b
f(x) = x^2 + ax + b
f
(
x
)
=
x
2
+
a
x
+
b
and
g
(
x
)
=
x
2
+
c
x
+
d
g(x) = x^2 + cx + d
g
(
x
)
=
x
2
+
c
x
+
d
which satisfies
f
(
20
)
+
f
(
15
)
=
g
(
20
)
+
g
(
15
)
f(20) + f(15) = g(20) + g(15)
f
(
20
)
+
f
(
15
)
=
g
(
20
)
+
g
(
15
)
. The sum of all real numbers
x
x
x
that satisfy
f
(
x
)
=
g
(
x
)
f(x) = g(x)
f
(
x
)
=
g
(
x
)
is equal to ... p10. Given a and b positive integers with
53
201
<
a
b
<
4
15
\frac{53}{201} <\frac{a}{b}<\frac{4}{15}
201
53
<
b
a
<
15
4
. The smallest possible value of
b
b
b
is ... p11. Suppose in a laboratory there are
20
20
20
computers and
15
15
15
printers. Cables are used to connect computers and printers. Sadly, one printer can only serve one computer at a time together. Desired
15
15
15
computers can always use the printer on same time. The number of cables required to connect at least as many computers and printers is ... p12. Given a triangle
A
B
C
ABC
A
BC
with
M
M
M
in the midpoint of
B
C
BC
BC
, and on the side AB selected point
N
N
N
so that
N
B
=
2
N
A
NB = 2NA
NB
=
2
N
A
. If
∠
C
A
B
=
∠
C
M
N
\angle CAB = \angle CMN
∠
C
A
B
=
∠
CMN
, then the value of
A
C
B
C
\frac{AC}{BC}
BC
A
C
is ... p13. Given the sequence
a
0
,
a
1
,
a
2
,
.
.
.
a_0, a_1, a_2, ...
a
0
,
a
1
,
a
2
,
...
with
a
0
=
2
a_0 = 2
a
0
=
2
,
a
1
=
8
3
a_1 =\frac83
a
1
=
3
8
and
a
m
a
n
=
a
m
+
n
−
a
m
−
n
a_ma_n= a_{m+n}- a_{m-n}
a
m
a
n
=
a
m
+
n
−
a
m
−
n
for every natural number
m
,
n
m, n
m
,
n
with
m
≥
n
m\ge n
m
≥
n
. The number of natural numbers
n
n
n
that fulfills
a
n
−
3
n
>
1
2015
a_n-3^n > \frac{1}{2015}
a
n
−
3
n
>
2015
1
is ... p14. For the real number x, the notation
⌊
x
⌋
\lfloor x \rfloor
⌊
x
⌋
denotes the largest integer that not greater than
x
x
x
, while
⌈
x
⌉
\lceil x \rceil
⌈
x
⌉
represents the smallest integer which is not smaller than
x
x
x
. The real number
x
x
x
that satisfies
⌊
x
2
⌋
−
3
x
+
⌈
x
⌉
=
0
\lfloor x^2 \rfloor -3x + \lceil x \rceil = 0
⌊
x
2
⌋
−
3
x
+
⌈
x
⌉
=
0
is ... p15. A circle intersects an equilateral triangle
A
B
C
ABC
A
BC
at six points that are different. About the six intersection points, every two points located on a different side of the triangle, so that:
B
,
D
,
E
,
C
B,D,E,C
B
,
D
,
E
,
C
and
C
,
F
,
G
,
A
C, F, G,A
C
,
F
,
G
,
A
, and
A
,
H
,
J
,
B
A,H,J,B
A
,
H
,
J
,
B
are in a row in a line. If
A
G
=
2
AG = 2
A
G
=
2
,
G
F
=
13
GF = 13
GF
=
13
,
F
C
=
1
FC = 1
FC
=
1
, and
H
J
=
7
HJ = 7
H
J
=
7
, then the length of
D
E
DE
D
E
is ... p16. In the picture there are as many triangles as ...... https://cdn.artofproblemsolving.com/attachments/b/b/bfd55c68a906f7b4c41ffa07728f0602f2afc1.png p17. Let
M
M
M
and
m
m
m
be the largest and smallest values of a, respectively such that
∣
x
2
−
2
a
x
−
a
2
−
3
4
∣
≤
1
\left|x^2-2ax-a^2-\frac34 \right| \le 1
x
2
−
2
a
x
−
a
2
−
4
3
≤
1
for every
x
∈
[
0
,
1
]
x\in[0, 1]
x
∈
[
0
,
1
]
. The value of
M
−
m
M-m
M
−
m
is ... p18. All integers
n
n
n
such that
9
n
+
1
n
+
3
\frac{9n + 1}{n + 3}
n
+
3
9
n
+
1
is the square of a rational number are ... p19. The set
A
A
A
, subset of
{
1
,
2
,
.
.
.
,
15
}
\{1, 2,..., 15\}
{
1
,
2
,
...
,
15
}
is said to be good, if for every
a
∈
A
a \in A
a
∈
A
applies
a
−
1
∈
A
a-1 \in A
a
−
1
∈
A
or
a
+
1
∈
A
a + 1 \in A
a
+
1
∈
A
. The number of good subsets with five elements of of
{
1
,
2
,
.
.
.
,
15
}
\{1, 2,..., 15\}
{
1
,
2
,
...
,
15
}
is ... p20. Given an isosceles triangle
A
B
C
ABC
A
BC
, where
A
B
=
A
C
=
b
AB = AC = b
A
B
=
A
C
=
b
,
B
C
=
a
BC = a
BC
=
a
, and
∠
B
A
C
=
10
0
o
\angle BAC = 100^o
∠
B
A
C
=
10
0
o
. If
B
L
BL
B
L
bisects
∠
A
B
C
\angle ABC
∠
A
BC
, then the value of
A
L
+
B
L
AL + BL
A
L
+
B
L
is ...