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National and Regional Contests
Indonesia Contests
Indonesia MO Shortlist
2015 Indonesia MO Shortlist
2015 Indonesia MO Shortlist
Part of
Indonesia MO Shortlist
Subcontests
(31)
C2
1
Hide problems
dividing a set into 2 sets with same sum each
Given
2
n
2n
2
n
natural numbers, so that the average arithmetic of those
2
n
2n
2
n
number is
2
2
2
. If all the number is not more than
2
n
2n
2
n
. Prove we can divide those
2
n
2n
2
n
numbers into
2
2
2
sets, so that the sum of each set to be the same.
C6
1
Hide problems
fleas on the real line jumping over integers
Let
k
k
k
be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance
k
k
k
is the original distance. Formally, we may choose
2
2
2
tails
A
,
B
A, B
A
,
B
that are spaced
n
n
n
and move
A
A
A
to the different side of
B
B
B
so the current distance is
k
n
kn
kn
. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of
k
k
k
so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.
G2
1
Hide problems
externally tangent circles, perpendicular chords, collinearity wanted
Two circles that are not equal are tangent externally at point
R
R
R
. Suppose point
P
P
P
is the intersection of the external common tangents of the two circles. Let
A
A
A
and
B
B
B
are two points on different circles so that
R
A
RA
R
A
is perpendicular to
R
B
RB
RB
. Show that the line
A
B
AB
A
B
passes through
P
P
P
.
G4
1
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equal segments wanted, related to an isosceles triangle and 2 circumcircles
Given an isosceles triangle
A
B
C
ABC
A
BC
with
A
B
=
A
C
AB = AC
A
B
=
A
C
, suppose
D
D
D
is the midpoint of the
A
C
AC
A
C
. The circumcircle of the
D
B
C
DBC
D
BC
triangle intersects the altitude from
A
A
A
at point
E
E
E
inside the triangle
A
B
C
ABC
A
BC
, and the circumcircle of the triangle
A
E
B
AEB
A
EB
cuts the side
B
D
BD
B
D
at point
F
F
F
. If
C
F
CF
CF
cuts
A
E
AE
A
E
at point
G
G
G
, prove that
A
E
=
E
G
AE = EG
A
E
=
EG
.
G3
1
Hide problems
from 3 points of circumcircle, two tangents to incircle, means one more
Given
A
B
C
ABC
A
BC
triangle with incircle
L
1
L_1
L
1
and circumcircle
L
2
L_2
L
2
. If points
X
,
Y
,
Z
X, Y, Z
X
,
Y
,
Z
lie on
L
2
L_2
L
2
, such that
X
Y
,
X
Z
XY, XZ
X
Y
,
XZ
are tangent to
L
1
L_1
L
1
, then prove that
Y
Z
YZ
Y
Z
is also tangent to
L
1
L_1
L
1
.
G5
1
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if A, P, Q are collinear, then AB = AC , two circles related
Let
A
B
C
ABC
A
BC
be an acute triangle. Suppose that circle
Γ
1
\Gamma_1
Γ
1
has it's center on the side
A
C
AC
A
C
and is tangent to the sides
A
B
AB
A
B
and
B
C
BC
BC
, and circle
Γ
2
\Gamma_2
Γ
2
has it's center on the side
A
B
AB
A
B
and is tangent to the sides
A
C
AC
A
C
and
B
C
BC
BC
. The circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at two points
P
P
P
and
Q
Q
Q
. Show that if
A
,
P
,
Q
A, P, Q
A
,
P
,
Q
are collinear, then
A
B
=
A
C
AB = AC
A
B
=
A
C
.
C7
1
Hide problems
subset of {1,2, 3,... , 2014}, with 12 elements, with numbers of sum = 2015
Show that there is a subset of
A
A
A
from
{
1
,
2
,
3
,
.
.
.
,
2014
}
\{1,2, 3,... , 2014\}
{
1
,
2
,
3
,
...
,
2014
}
such that : (i)
∣
A
∣
=
12
|A| = 12
∣
A
∣
=
12
(ii) for each coloring number in
A
A
A
with red or white , we can always find some numbers colored in
A
A
A
whose sum is
2015
2015
2015
.
C5
1
Hide problems
max number of ways to choose representatives to speak
A meeting was attended by
n
n
n
people. They are welcome to occupy the
k
k
k
table provided
(
k
≤
n
2
)
\left( k \le \frac{n}{2} \right)
(
k
≤
2
n
)
. Each table is occupied by at least two people. When the meeting begins, the moderator selects two people from each table as representatives for talk to. Suppose that
A
A
A
is the number of ways to choose representatives to speak. Determine the maximum value of
A
A
A
that is possible.
C3
1
Hide problems
2015 marbles in a box, 3 colors at start, one color in the end
We have
2015
2015
2015
marbles in a box, where each marble has one color from red, green or blue. At each step, we are allowed to take
2
2
2
different colored marbles, then replace it with
2
2
2
marbles with the third color. For example, we take one blue marble and one green marble, and we fill with
2
2
2
red marbles. Prove that we can always do a series of steps so that all marbles in the box have the same color.
C1
1
Hide problems
2015 elephants on a chessboard of dimensions 2x 2015 without threats
Given natural number n. Suppose that
N
N
N
is the maximum number of elephants that can be placed on a chessboard measuring
2
×
n
2 \times n
2
×
n
so that no two elephants are mutually under attack. Determine the number of ways to put
N
N
N
elephants on a chessboard sized
2
×
n
2 \times n
2
×
n
so that no two elephants attack each other.Alternative Formulation: Determine the number of ways to put
2015
2015
2015
elephants on a chessboard measuring
2
×
2015
2 \times 2015
2
×
2015
so there are no two elephants attacking each othePS. Elephant = Bishop
G8
1
Hide problems
circumcircle tangent to given circle
A
B
C
ABC
A
BC
is an acute triangle with
A
B
>
A
C
AB> AC
A
B
>
A
C
.
Γ
B
\Gamma_B
Γ
B
is a circle that passes through
A
,
B
A,B
A
,
B
and is tangent to
A
C
AC
A
C
on
A
A
A
. Define similar for
Γ
C
\Gamma_C
Γ
C
. Let
D
D
D
be the intersection
Γ
B
\Gamma_B
Γ
B
and
Γ
C
\Gamma_C
Γ
C
and
M
M
M
be the midpoint of
B
C
BC
BC
.
A
M
AM
A
M
cuts
Γ
C
\Gamma_C
Γ
C
at
E
E
E
. Let
O
O
O
be the center of the circumscibed circle of the triangle
A
B
C
ABC
A
BC
. Prove that the circumscibed circle of the triangle
O
D
E
ODE
O
D
E
is tangent to
Γ
B
\Gamma_B
Γ
B
.
N8
1
Hide problems
infinite n such that exists naturals a,b: a + b = n and ab | n^2 + n + 1
The natural number
n
n
n
is said to be good if there are natural numbers
a
a
a
and
b
b
b
that satisfy
a
+
b
=
n
a + b = n
a
+
b
=
n
and
a
b
∣
n
2
+
n
+
1
ab | n^2 + n + 1
ab
∣
n
2
+
n
+
1
. (a) Show that there are infinitely many good numbers. (b) Show that if
n
n
n
is a good number, then
7
∤
n
7 \nmid n
7
∤
n
.
N6
1
Hide problems
prove that each beautiful set must be charming (in non negative integers)
Defined as
N
0
N_0
N
0
as the set of all non-negative integers. Set
S
⊂
N
0
S \subset N_0
S
⊂
N
0
with not so many elements is called beautiful if for every
a
,
b
∈
S
a, b \in S
a
,
b
∈
S
with
a
≥
b
a \ge b
a
≥
b
(
a
a
a
and
b
b
b
do not have to be different), exactly one of
a
+
b
a + b
a
+
b
or
a
−
b
a - b
a
−
b
is in
S
S
S
. Set
T
⊂
N
0
T \subset N_0
T
⊂
N
0
with not so many elements is called charming if the largest number
k
k
k
such that up to 3
k
∣
a
^k | a
k
∣
a
is the same for each element
a
∈
T
a \in T
a
∈
T
. Prove that each beautiful set must be charming.
N5
1
Hide problems
ax+by=n, odd no. of non-negative integer solutions (x, y)
Given a prime number
n
≥
5
n \ge 5
n
≥
5
. Prove that for any natural number
a
≤
n
2
a \le \frac{n}{2}
a
≤
2
n
, we can search for natural number
b
≤
n
2
b \le \frac{n}{2}
b
≤
2
n
so the number of non-negative integer solutions
(
x
,
y
)
(x, y)
(
x
,
y
)
of the equation
a
x
+
b
y
=
n
ax+by=n
a
x
+
b
y
=
n
to be odd*.Clarification: * For example when
n
=
7
,
a
=
3
n = 7, a = 3
n
=
7
,
a
=
3
, we can choose
b
=
1
b = 1
b
=
1
so that there number of solutions og
3
x
+
y
=
7
3x + y = 7
3
x
+
y
=
7
to be
3
3
3
(odd), namely:
(
0
,
7
)
,
(
1
,
4
)
,
(
2
,
1
)
(0, 7), (1, 4), (2, 1)
(
0
,
7
)
,
(
1
,
4
)
,
(
2
,
1
)
N4
1
Hide problems
a^ab^{a + b} = c^cd^{c + d}, gcd (a, b) = gcd $(c, d) = 1=>(a,b)=(c,d)
Suppose that the natural number
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
satisfy the equation
a
a
b
a
+
b
=
c
c
d
c
+
d
a^ab^{a + b} = c^cd^{c + d}
a
a
b
a
+
b
=
c
c
d
c
+
d
. (a) If gcd
(
a
,
b
)
=
(a, b) =
(
a
,
b
)
=
gcd
(
c
,
d
)
=
1
(c, d) = 1
(
c
,
d
)
=
1
, prove that
a
=
c
a = c
a
=
c
and
b
=
d
b = d
b
=
d
. (b) Does the conclusion
a
=
c
a = c
a
=
c
and
b
=
d
b = d
b
=
d
apply, without the condition gcd
(
a
,
b
)
=
(a, b) =
(
a
,
b
)
=
gcd
(
c
,
d
)
=
1
(c, d) = 1
(
c
,
d
)
=
1
?
N2
1
Hide problems
a,b in N: x^2+ax-b , x^2-ax+b have integer roots, existence of a right triangle
Suppose that
a
,
b
a, b
a
,
b
are natural numbers so that all the roots of
x
2
+
a
x
−
b
x^2 + ax - b
x
2
+
a
x
−
b
and
x
2
−
a
x
+
b
x^2 - ax + b
x
2
−
a
x
+
b
are integers. Show that exists a right triangle with integer sides, with
a
a
a
the length of the hypotenuse and
b
b
b
the area .
N1
1
Hide problems
min of abc when a>b>b, primes, a=b+2c, a + b + c is perfect square
A triple integer
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is called brilliant when it satisfies: (i)
a
>
b
>
c
a> b> c
a
>
b
>
c
are prime numbers (ii)
a
=
b
+
2
c
a = b + 2c
a
=
b
+
2
c
(iii)
a
+
b
+
c
a + b + c
a
+
b
+
c
is a perfect square number Find the minimum value of
a
b
c
abc
ab
c
if triple
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
is brilliant.
G1
1
Hide problems
rectangle ABCD so that AB = AD and AB + BC <CD, prove <ABC >120^o
Given a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
so that
A
B
=
A
D
AB = AD
A
B
=
A
D
and
A
B
+
B
C
<
C
D
AB + BC <CD
A
B
+
BC
<
C
D
. Prove that the angle
A
B
C
ABC
A
BC
is more than
120
120
120
degrees.
A2
1
Hide problems
\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}, prove a is a natural
Suppose
a
a
a
real number so that there is a non-constant polynomial
P
(
x
)
P (x)
P
(
x
)
such that
P
(
x
+
1
)
−
P
(
x
)
P
(
x
+
π
)
=
a
x
+
π
\frac{P(x+1)-P(x)}{P(x+\pi)}= \frac{a}{x+\pi}
P
(
x
+
π
)
P
(
x
+
1
)
−
P
(
x
)
=
x
+
π
a
for each real number
x
x
x
, with
x
+
π
≠
0
x+\pi \ne 0
x
+
π
=
0
and
P
(
x
+
π
)
≠
0
P(x+\pi)\ne 0
P
(
x
+
π
)
=
0
. Show that
a
a
a
is a natural number.
A1
1
Hide problems
f_1 (x) f_2 (x) is periodic when f_1 (x), f_2 (x) are periodic
Function
f
:
R
→
R
f: R\to R
f
:
R
→
R
is said periodic , if
f
f
f
is not a constant function and there is a number real positive
p
p
p
with the property of
f
(
x
)
=
f
(
x
+
p
)
f (x) = f (x + p)
f
(
x
)
=
f
(
x
+
p
)
for every
x
∈
R
x \in R
x
∈
R
. The smallest positive real number p which satisfies the condition
f
(
x
)
=
f
(
x
+
p
)
f (x) = f (x + p)
f
(
x
)
=
f
(
x
+
p
)
for each
x
∈
R
x \in R
x
∈
R
is named period of
f
f
f
. Given
a
a
a
and
b
b
b
real positive numbers, show that there are periodic functions
f
1
f_1
f
1
and
f
2
f_2
f
2
, with periods
a
a
a
and
b
b
b
respectively, so that
f
1
(
x
)
⋅
f
2
(
x
)
f_1 (x)\cdot f_2 (x)
f
1
(
x
)
⋅
f
2
(
x
)
is also a periodic function.
A3
1
Hide problems
Inequality....
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
positive reals such that
a
2
+
b
2
+
c
2
=
1
a^2+b^2+c^2=1
a
2
+
b
2
+
c
2
=
1
. Prove that
a
+
b
a
b
+
1
+
b
+
c
b
c
+
1
+
c
+
a
a
c
+
1
≤
3
\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3
ab
+
1
a
+
b
+
b
c
+
1
b
+
c
+
a
c
+
1
c
+
a
≤
3
A7
1
Hide problems
Integer Polynomials!
Suppose
P
(
n
)
P(n)
P
(
n
)
is a nonconstant polynomial where all of its coefficients are nonnegative integers such that
∑
i
=
1
n
P
(
i
)
∣
n
P
(
n
+
1
)
\sum_{i=1}^n P(i) | nP(n+1)
i
=
1
∑
n
P
(
i
)
∣
n
P
(
n
+
1
)
for every
n
∈
N
n \in \mathbb{N}
n
∈
N
. Prove that there exists an integer
k
≥
0
k \ge 0
k
≥
0
such that
P
(
n
)
=
(
n
+
k
n
−
1
)
P
(
1
)
P(n) = \binom{n+k}{n-1} P(1)
P
(
n
)
=
(
n
−
1
n
+
k
)
P
(
1
)
for every
n
∈
N
n \in \mathbb{N}
n
∈
N
.
C9
1
Hide problems
A game but is this already set up?
Given 2015 balls. Astri and Budi will play a game. At first, Astri will choose two different numbers
a
a
a
and
b
b
b
from the set
S
=
{
1
,
2
,
3
,
…
,
30
}
S = \{ 1, 2, 3, \dots, 30 \}
S
=
{
1
,
2
,
3
,
…
,
30
}
. Budi will then choose another 2 different numbers
c
c
c
and
d
d
d
from the remaining 28 numbers in set
S
S
S
. By taking turns, starting from Astri, they take balls with the following rules: (1) Astri could only take
a
a
a
or
b
b
b
balls. (2) Budi could only take
c
c
c
or
d
d
d
balls. until someone couldn't take any balls satisfying the condition given (and that person will lose). Prove that Budi could choose
c
,
d
c,d
c
,
d
such that he has a strategy to ensure his victory on this game.
A4
1
Hide problems
Sorry, Just too many functions
Determine all functions
f
:
R
×
R
→
R
f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}
f
:
R
×
R
→
R
such that
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
max
{
x
,
y
,
z
}
−
min
{
x
,
y
,
z
}
f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \}
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
max
{
x
,
y
,
z
}
−
min
{
x
,
y
,
z
}
for every
x
,
y
,
z
∈
R
x,y,z \in \mathbb{R}
x
,
y
,
z
∈
R
and there exists some real
a
a
a
such that
f
(
x
,
a
)
=
f
(
a
,
x
)
f(x,a) = f(a,x)
f
(
x
,
a
)
=
f
(
a
,
x
)
for every
x
∈
R
x \in \mathbb{R}
x
∈
R
.
A6
1
Hide problems
I can't do two fe :")
Let functions
f
,
g
:
R
+
→
R
+
f, g: \mathbb{R}^+ \to \mathbb{R}^+
f
,
g
:
R
+
→
R
+
satisfy the following:
f
(
g
(
x
)
y
+
f
(
x
)
)
=
(
y
+
2015
)
f
(
x
)
f(g(x)y + f(x)) = (y+2015)f(x)
f
(
g
(
x
)
y
+
f
(
x
))
=
(
y
+
2015
)
f
(
x
)
for every
x
,
y
∈
R
+
x,y \in \mathbb{R}^+
x
,
y
∈
R
+
. (a) Prove that
g
(
x
)
=
f
(
x
)
2015
g(x) = \frac{f(x)}{2015}
g
(
x
)
=
2015
f
(
x
)
for every
x
∈
R
+
.
x \in \mathbb{R}^+.
x
∈
R
+
.
(b) State an example of function that satisfy the equation above and
f
(
x
)
,
g
(
x
)
≥
1
f(x), g(x) \ge 1
f
(
x
)
,
g
(
x
)
≥
1
for every
x
∈
R
+
x \in \mathbb{R}^+
x
∈
R
+
.
C8
1
Hide problems
colouring the floor's window of three 2025 floor buidings with 3 colors
It is known that there are
3
3
3
buildings in the same shape which are located in an equilateral triangle. Each building has a
2015
2015
2015
floor with each floor having one window. In all three buildings, every
1
1
1
st floor is uninhabited, while each floor of others have exactly one occupant. All windows will be colored with one of red, green or blue. The residents of each floor of a building can see the color of the window in the other buildings of the the same floor and one floor just below it, but they cannot see the colors of the other windows of the two buildings. Besides that, sresidents cannot see the color of the window from any floor in the building itself. For example, resident of the
10
10
10
th floor can see the colors of the
9
9
9
th and
10
10
10
th floor windows for the other buildings (a total of
4
4
4
windows) and he can't see the color of the other window. We want to color the windows so that each resident can see at lest
1
1
1
window of each color. How many ways are there to color those windows?
A5
1
Hide problems
sum \sqrt{\frac{a}{b+c}+\frac{b}{c+a}} >=3
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers. Prove that
a
b
+
c
+
b
c
+
a
+
b
c
+
a
+
c
a
+
b
+
c
a
+
b
+
a
b
+
c
≥
3
\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3
b
+
c
a
+
c
+
a
b
+
c
+
a
b
+
a
+
b
c
+
a
+
b
c
+
b
+
c
a
≥
3
G6
1
Hide problems
EQ = FQ iff BP = CP
Let
A
B
C
ABC
A
BC
be an acute angled triangle with circumcircle
O
O
O
. Line
A
O
AO
A
O
intersects the circumcircle of triangle
A
B
C
ABC
A
BC
again at point
D
D
D
. Let
P
P
P
be a point on the side
B
C
BC
BC
. Line passing through
P
P
P
perpendicular to
A
P
AP
A
P
intersects lines
D
B
DB
D
B
and
D
C
DC
D
C
at
E
E
E
and
F
F
F
respectively . Line passing through
D
D
D
perpendicular to
B
C
BC
BC
intersects
E
F
EF
EF
at point
Q
Q
Q
. Prove that
E
Q
=
F
Q
EQ = FQ
EQ
=
FQ
if and only if
B
P
=
C
P
BP = CP
BP
=
CP
.
N3
1
Hide problems
Divisibility of power of integer
Given positive integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that
a
∣
c
d
a\mid c^d
a
∣
c
d
and
b
∣
d
c
b\mid d^c
b
∣
d
c
. Prove that
a
b
∣
(
c
d
)
m
a
x
(
a
,
b
)
ab\mid (cd)^{max(a,b)}
ab
∣
(
c
d
)
ma
x
(
a
,
b
)
G7
1
Hide problems
Geometry Collinearity
Given an acute triangle
A
B
C
ABC
A
BC
.
Γ
B
\Gamma _{B}
Γ
B
is a circle that passes through
A
B
AB
A
B
, tangent to
A
C
AC
A
C
at
A
A
A
and centered at
O
B
O_{B}
O
B
. Define
Γ
C
\Gamma_C
Γ
C
and
O
C
O_C
O
C
the same way. Let the altitudes of
△
A
B
C
\triangle ABC
△
A
BC
from
B
B
B
and
C
C
C
meets the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
at
X
X
X
and
Y
Y
Y
, respectively. Prove that
A
A
A
, the midpoint of
X
Y
XY
X
Y
and the midpoint of
O
B
O
C
O_{B}O_{C}
O
B
O
C
is collinear.
N7
1
Hide problems
gcd and lcm
For every natural number
a
a
a
and
b
b
b
, define the notation
[
a
,
b
]
[a,b]
[
a
,
b
]
as the least common multiple of
a
a
a
and
b
b
b
and the notation
(
a
,
b
)
(a,b)
(
a
,
b
)
as the greatest common divisor of
a
a
a
and
b
b
b
. Find all
n
∈
N
n \in \mathbb{N}
n
∈
N
that satisfies
4
∑
k
=
1
n
[
n
,
k
]
=
1
+
∑
k
=
1
n
(
n
,
k
)
+
2
n
2
∑
k
=
1
n
1
(
n
,
k
)
4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}
4
k
=
1
∑
n
[
n
,
k
]
=
1
+
k
=
1
∑
n
(
n
,
k
)
+
2
n
2
k
=
1
∑
n
(
n
,
k
)
1