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Problems
Contests
National and Regional Contests
Indonesia Contests
Indonesia MO Shortlist
2014 Indonesia MO Shortlist
2014 Indonesia MO Shortlist
Part of
Indonesia MO Shortlist
Subcontests
(24)
G6
1
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tangent to the circumcircle when tangent to a given point and a line
Given an
A
B
C
ABC
A
BC
acute triangle with
O
O
O
the center of the circumscribed circle. Suppose that
ω
\omega
ω
is a circle that is tangent to the line
A
O
AO
A
O
at point
A
A
A
and also tangent to the line
B
C
BC
BC
. Prove that
ω
\omega
ω
is also tangent to the circumcircle of the triangle
B
O
C
BOC
BOC
.
G5
1
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equal angles wanted, 4 midpoints related
Given a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. Suppose
E
,
F
,
G
,
H
E, F, G, H
E
,
F
,
G
,
H
are respectively the midpoint of the sides
A
B
,
B
C
,
C
D
,
D
A
AB, BC, CD, DA
A
B
,
BC
,
C
D
,
D
A
. The line passing through
G
G
G
and perpendicular on
A
B
AB
A
B
intersects the line passing through
H
H
H
and perpendicular on
B
C
BC
BC
at point
K
K
K
. Prove that
∠
E
K
F
=
∠
A
B
C
\angle EKF = \angle ABC
∠
E
K
F
=
∠
A
BC
.
G4
1
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kite and angle bisector given, concurrency wanted
Given an acute triangle
A
B
C
ABC
A
BC
with
A
B
<
A
C
AB <AC
A
B
<
A
C
. Points
P
P
P
and
Q
Q
Q
lie on the angle bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
so that
B
P
BP
BP
and
C
Q
CQ
CQ
are perpendicular on that angle bisector. Suppose that point
E
,
F
E, F
E
,
F
lie respectively at sides
A
B
AB
A
B
and
A
C
AC
A
C
respectively, in such a way that
A
E
P
F
AEPF
A
EPF
is a kite. Prove that the lines
B
C
,
P
F
BC, PF
BC
,
PF
, and
Q
E
QE
QE
intersect at one point.
G1
1
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cyclic wanted, 3 reflections and incircle related
The inscribed circle of the
A
B
C
ABC
A
BC
triangle has center
I
I
I
and touches to
B
C
BC
BC
at
X
X
X
. Suppose the
A
I
AI
A
I
and
B
C
BC
BC
lines intersect at
L
L
L
, and
D
D
D
is the reflection of
L
L
L
wrt
X
X
X
. Points
E
E
E
and
F
F
F
respectively are the result of a reflection of
D
D
D
wrt to lines
C
I
CI
C
I
and
B
I
BI
B
I
respectively. Show that quadrilateral
B
C
E
F
BCEF
BCEF
is cyclic .
C6
1
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numbers in circle, circular arc with sum s, 1<= s <= 1/2 n(n+1)
Determine all natural numbers
n
n
n
so that numbers
1
,
2
,
.
.
.
,
n
1, 2,... , n
1
,
2
,
...
,
n
can be placed on the circumference of a circle and for each natural number
s
s
s
with
1
≤
s
≤
1
2
n
(
n
+
1
)
1\le s \le \frac12n(n+1)
1
≤
s
≤
2
1
n
(
n
+
1
)
, there is a circular arc which has the sum of all numbers in that arc to be
s
s
s
.
C5
1
Hide problems
\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r}
Determine all pairs of natural numbers
(
m
,
r
)
(m, r)
(
m
,
r
)
with
2014
≥
m
≥
r
≥
1
2014 \ge m \ge r \ge 1
2014
≥
m
≥
r
≥
1
that fulfill
(
2014
m
)
+
(
m
r
)
=
(
2014
r
)
+
(
2014
−
r
m
−
r
)
\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r}
(
m
2014
)
+
(
r
m
)
=
(
r
2014
)
+
(
m
−
r
2014
−
r
)
C3
1
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coloring with 2 colours a mxn chessboard iff m \cdot n is even
Let
n
n
n
be a natural number. Given a chessboard sized
m
×
n
m \times n
m
×
n
. The sides of the small squares of chessboard are not on the perimeter of the chessboard will be colored so that each small square has exactly two sides colored. Prove that a coloring like that is possible if and only if
m
⋅
n
m \cdot n
m
⋅
n
is even.
C2
1
Hide problems
min no to colour 1,2,...,2013 such a,b same color & ab not multiple of 2014
Show that the smallest number of colors that is needed for coloring numbers
1
,
2
,
.
.
.
,
2013
1, 2,..., 2013
1
,
2
,
...
,
2013
so that for every two number
a
,
b
a, b
a
,
b
which is the same color,
a
b
ab
ab
is not a multiple of
2014
2014
2014
, is
3
3
3
colors.
N1
1
Hide problems
ab + (a + 1) (b + 1) = 2^k diophantine and k+1 prime
(a) Let
k
k
k
be an natural number so that the equation
a
b
+
(
a
+
1
)
(
b
+
1
)
=
2
k
ab + (a + 1) (b + 1) = 2^k
ab
+
(
a
+
1
)
(
b
+
1
)
=
2
k
does not have a positive integer solution
(
a
,
b
)
(a, b)
(
a
,
b
)
. Show that
k
+
1
k + 1
k
+
1
is a prime number. (b) Show that there are natural numbers
k
k
k
so that
k
+
1
k + 1
k
+
1
is prime numbers and equation
a
b
+
(
a
+
1
)
(
b
+
1
)
=
2
k
ab + (a + 1) (b + 1) = 2^k
ab
+
(
a
+
1
)
(
b
+
1
)
=
2
k
has a positive integer solution
(
a
,
b
)
(a, b)
(
a
,
b
)
.
N2
1
Hide problems
if abc = k^2 + 1 in N, then one of a - 1, b - 1, c - 1 is a composite
Suppose that
a
,
b
,
c
,
k
a, b, c, k
a
,
b
,
c
,
k
are natural numbers with
a
,
b
,
c
≥
3
a, b, c \ge 3
a
,
b
,
c
≥
3
which fulfill the equation
a
b
c
=
k
2
+
1
abc = k^2 + 1
ab
c
=
k
2
+
1
. Show that at least one between
a
−
1
,
b
−
1
,
c
−
1
a - 1, b - 1, c -1
a
−
1
,
b
−
1
,
c
−
1
is composite number.
N3
1
Hide problems
a^b=(a+b)^a in NxN
Find all pairs of natural numbers
(
a
,
b
)
(a, b)
(
a
,
b
)
that fulfill
a
b
=
(
a
+
b
)
a
a^b=(a+b)^a
a
b
=
(
a
+
b
)
a
.
A1
1
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\lfloor a^{2014} \rfloor + \lfloor b^{2014} \rfloor =\lfloor a \rfloor ^{2014}
Let
a
,
b
a, b
a
,
b
be positive real numbers such that there exist infinite number of natural numbers
k
k
k
such that
⌊
a
k
⌋
+
⌊
b
k
⌋
=
⌊
a
⌋
k
+
⌊
b
⌋
k
\lfloor a^k \rfloor + \lfloor b^k \rfloor = \lfloor a \rfloor ^k + \lfloor b \rfloor ^k
⌊
a
k
⌋
+
⌊
b
k
⌋
=
⌊
a
⌋
k
+
⌊
b
⌋
k
. Prove that
⌊
a
2014
⌋
+
⌊
b
2014
⌋
=
⌊
a
⌋
2014
+
⌊
b
⌋
2014
\lfloor a^{2014} \rfloor + \lfloor b^{2014} \rfloor = \lfloor a \rfloor ^{2014} + \lfloor b \rfloor ^{2014}
⌊
a
2014
⌋
+
⌊
b
2014
⌋
=
⌊
a
⌋
2014
+
⌊
b
⌋
2014
A5
1
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(a_1+a_2+...+a_m)/m >= \sqrt{(a_1^2+a_2^2+...+a_{2014}^2)/2014} , max m
Determine the largest natural number
m
m
m
such that for each non negative real numbers
a
1
≥
a
2
≥
.
.
.
≥
a
2014
≥
0
a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0
a
1
≥
a
2
≥
...
≥
a
2014
≥
0
, it is true that
a
1
+
a
2
+
.
.
.
+
a
m
m
≥
a
1
2
+
a
2
2
+
.
.
.
+
a
2014
2
2014
\frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}}
m
a
1
+
a
2
+
...
+
a
m
≥
2014
a
1
2
+
a
2
2
+
...
+
a
2014
2
A4
1
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(a+b+c)/5 <= \sqrt[3]{abc} if 1 <= a, b, c <= 8
Prove that for every real positive number
a
,
b
,
c
a, b, c
a
,
b
,
c
with
1
≤
a
,
b
,
c
≤
8
1 \le a, b, c \le 8
1
≤
a
,
b
,
c
≤
8
the inequality
a
+
b
+
c
5
≤
a
b
c
3
\frac{a+b+c}{5}\le \sqrt[3]{abc}
5
a
+
b
+
c
≤
3
ab
c
A3
1
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x^2y/(x+2y)+y^2z/(y+2z)+z^2x/(z+2x)<(x+y+z)^2/8 , in positive real
Prove for each positive real number
x
,
y
,
z
x, y, z
x
,
y
,
z
,
x
2
y
x
+
2
y
+
y
2
z
y
+
2
z
+
z
2
x
z
+
2
x
<
(
x
+
y
+
z
)
2
8
\frac{x^2y}{x+2y}+\frac{y^2z}{y+2z}+\frac{z^2x}{z+2x}<\frac{(x+y+z)^2}{8}
x
+
2
y
x
2
y
+
y
+
2
z
y
2
z
+
z
+
2
x
z
2
x
<
8
(
x
+
y
+
z
)
2
N5
1
Hide problems
N5 Indonesian 2014 Shortlist
Prove that we can give a color to each of the numbers
1
,
2
,
3
,
.
.
.
,
2013
1,2,3,...,2013
1
,
2
,
3
,
...
,
2013
with seven distinct colors (all colors are necessarily used) such that for any distinct numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
of the same color, then
2014
∤
a
b
c
2014\nmid abc
2014
∤
ab
c
and the remainder when
a
b
c
abc
ab
c
is divided by
2014
2014
2014
is of the same color as
a
,
b
,
c
a,b,c
a
,
b
,
c
.
N6
1
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Two ugly numbers don't make them beautiful
A positive integer is called beautiful if it can be represented in the form
x
2
+
y
2
x
+
y
\dfrac{x^2+y^2}{x+y}
x
+
y
x
2
+
y
2
for two distinct positive integers
x
,
y
x,y
x
,
y
. A positive integer that is not beautiful is ugly.a) Prove that
2014
2014
2014
is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.
C4
1
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Binomial coefficients everywhere
Suppose that
k
,
m
,
n
k,m,n
k
,
m
,
n
are positive integers with
k
≤
n
k \le n
k
≤
n
. Prove that:
∑
r
=
0
m
k
(
m
r
)
(
n
k
)
(
r
+
k
)
(
m
+
n
r
+
k
)
=
1
\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1
r
=
0
∑
m
(
r
+
k
)
(
r
+
k
m
+
n
)
k
(
r
m
)
(
k
n
)
=
1
G2
1
Hide problems
Going to the third power
Let
A
B
C
ABC
A
BC
be a triangle. Suppose
D
D
D
is on
B
C
BC
BC
such that
A
D
AD
A
D
bisects
∠
B
A
C
\angle BAC
∠
B
A
C
. Suppose
M
M
M
is on
A
B
AB
A
B
such that
∠
M
D
A
=
∠
A
B
C
\angle MDA = \angle ABC
∠
M
D
A
=
∠
A
BC
, and
N
N
N
is on
A
C
AC
A
C
such that
∠
N
D
A
=
∠
A
C
B
\angle NDA = \angle ACB
∠
N
D
A
=
∠
A
CB
. If
A
D
AD
A
D
and
M
N
MN
MN
intersect on
P
P
P
, prove that
A
D
3
=
A
B
⋅
A
C
⋅
A
P
AD^3 = AB \cdot AC \cdot AP
A
D
3
=
A
B
⋅
A
C
⋅
A
P
.
A2
1
Hide problems
Equal sum elements for equal product indices
A sequence of positive integers
a
1
,
a
2
,
…
a_1, a_2, \ldots
a
1
,
a
2
,
…
satisfies
a
k
+
a
l
=
a
m
+
a
n
a_k + a_l = a_m + a_n
a
k
+
a
l
=
a
m
+
a
n
for all positive integers
k
,
l
,
m
,
n
k,l,m,n
k
,
l
,
m
,
n
satisfying
k
l
=
m
n
kl = mn
k
l
=
mn
. Prove that if
p
p
p
divides
q
q
q
then
a
p
≤
a
q
a_p \le a_q
a
p
≤
a
q
.
A6
1
Hide problems
Polynomials forming right-angled triangles
Determine all polynomials with integral coefficients
P
(
x
)
P(x)
P
(
x
)
such that if
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a right-angled triangle, then
P
(
a
)
,
P
(
b
)
,
P
(
c
)
P(a), P(b), P(c)
P
(
a
)
,
P
(
b
)
,
P
(
c
)
are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if
c
c
c
is the hypotenuse of the first triangle, it's not necessary that
P
(
c
)
P(c)
P
(
c
)
is the hypotenuse of the second triangle, and similar with the others.)
G3
1
Hide problems
Playing with trapezoids
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid (quadrilateral with one pair of parallel sides) such that
A
B
<
C
D
AB < CD
A
B
<
C
D
. Suppose that
A
C
AC
A
C
and
B
D
BD
B
D
meet at
E
E
E
and
A
D
AD
A
D
and
B
C
BC
BC
meet at
F
F
F
. Construct the parallelograms
A
E
D
K
AEDK
A
E
DK
and
B
E
C
L
BECL
BEC
L
. Prove that
E
F
EF
EF
passes through the midpoint of the segment
K
L
KL
K
L
.
N4
1
Hide problems
Exactly one solution
For some positive integers
m
,
n
m,n
m
,
n
, the system
x
+
y
2
=
m
x+y^2 = m
x
+
y
2
=
m
and
x
2
+
y
=
n
x^2+y = n
x
2
+
y
=
n
has exactly one integral solution
(
x
,
y
)
(x,y)
(
x
,
y
)
. Determine all possible values of
m
−
n
m-n
m
−
n
.
C1
1
Hide problems
Adjacents sum to primes
Is it possible to fill a
3
×
3
3 \times 3
3
×
3
grid with each of the numbers
1
,
2
,
…
,
9
1,2,\ldots,9
1
,
2
,
…
,
9
once each such that the sum of any two numbers sharing a side is prime?