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Contests
International Contests
Balkan MO Shortlist
2014 Balkan MO Shortlist
2014 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(20)
C1
1
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no 4 brings bad luck, max no to be seated in a row of capacity 55 seats
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number
4
4
4
brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of
55
55
55
seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are
4
4
4
seats apart (so that bad luck during the competition is avoided)?
N5
1
Hide problems
BMO 2014 SL N5
N
5
\boxed{N5}
N
5
Let
a
,
b
,
c
,
p
,
q
,
r
a,b,c,p,q,r
a
,
b
,
c
,
p
,
q
,
r
be positive integers such that
a
p
+
b
q
+
c
r
=
a
q
+
b
r
+
c
p
=
a
r
+
b
p
+
c
q
.
a^p+b^q+c^r=a^q+b^r+c^p=a^r+b^p+c^q.
a
p
+
b
q
+
c
r
=
a
q
+
b
r
+
c
p
=
a
r
+
b
p
+
c
q
.
Prove that
a
=
b
=
c
a=b=c
a
=
b
=
c
or
p
=
q
=
r
.
p=q=r.
p
=
q
=
r
.
N3
1
Hide problems
BMO 2014 SL N3
N
3
\boxed{N3}
N
3
Prove that there exist infinitely many non isosceles triangles with rational side lengths
,
,
,
rational lentghs of altitudes and
,
,
,
perimeter equal to
3.
3.
3.
N2
1
Hide problems
BMO 2014 SL N2
N
2
\boxed{N2}
N
2
Let
p
p
p
be a prime numbers and
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
be integers.Show that if
x
1
n
+
x
2
n
+
.
.
.
+
x
p
n
≡
0
(
m
o
d
p
)
x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}
x
1
n
+
x
2
n
+
...
+
x
p
n
≡
0
(
mod
p
)
for all positive integers n then
x
1
≡
x
2
≡
.
.
.
≡
x
p
(
m
o
d
p
)
.
x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.
x
1
≡
x
2
≡
...
≡
x
p
(
mod
p
)
.
N1
1
Hide problems
BMO 2014 SL N1
N
1
\boxed{N1}
N
1
Let
n
n
n
be a positive integer,
g
(
n
)
g(n)
g
(
n
)
be the number of positive divisors of
n
n
n
of the form
6
k
+
1
6k+1
6
k
+
1
and
h
(
n
)
h(n)
h
(
n
)
be the number of positive divisors of
n
n
n
of the form
6
k
−
1
,
6k-1,
6
k
−
1
,
where
k
k
k
is a nonnegative integer.Find all positive integers
n
n
n
such that
g
(
n
)
g(n)
g
(
n
)
and
h
(
n
)
h(n)
h
(
n
)
have different parity.
A7
1
Hide problems
BMO 2014 SL A7
A
7
\boxed{A7}
A
7
Prove that for all
x
,
y
,
z
>
0
x,y,z>0
x
,
y
,
z
>
0
with
1
x
+
1
y
+
1
z
=
1
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1
x
1
+
y
1
+
z
1
=
1
and
0
≤
a
,
b
,
c
<
1
0\leq a,b,c<1
0
≤
a
,
b
,
c
<
1
the following inequality holds
x
2
+
y
2
1
−
a
z
+
y
2
+
z
2
1
−
b
x
+
z
2
+
x
2
1
−
c
y
≥
6
(
x
+
y
+
z
)
1
−
a
b
c
\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}
1
−
a
z
x
2
+
y
2
+
1
−
b
x
y
2
+
z
2
+
1
−
c
y
z
2
+
x
2
≥
1
−
ab
c
6
(
x
+
y
+
z
)
A6
1
Hide problems
BMO 2014 SL A6
A
6
\boxed{A6}
A
6
The sequence
a
0
,
a
1
,
.
.
.
a_0,a_1,...
a
0
,
a
1
,
...
is defined by the initial conditions
a
0
=
1
,
a
1
=
6
a_0=1,a_1=6
a
0
=
1
,
a
1
=
6
and the recursion
a
n
+
1
=
4
a
n
−
a
n
−
1
+
2
a_{n+1}=4a_n-a_{n-1}+2
a
n
+
1
=
4
a
n
−
a
n
−
1
+
2
for
n
>
1.
n>1.
n
>
1.
Prove that
a
2
k
−
1
a_{2^k-1}
a
2
k
−
1
has at least three prime factors for every positive integer
k
>
3.
k>3.
k
>
3.
A5
1
Hide problems
BMO 2014 SL A5
A
5
\boxed{A5}
A
5
Let
n
∈
N
,
n
>
2
n\in{N},n>2
n
∈
N
,
n
>
2
,and suppose
a
1
,
a
2
,
.
.
.
,
a
2
n
a_1,a_2,...,a_{2n}
a
1
,
a
2
,
...
,
a
2
n
is a permutation of the numbers
1
,
2
,
.
.
.
,
2
n
1,2,...,2n
1
,
2
,
...
,
2
n
such that
a
1
<
a
3
<
.
.
.
<
a
2
n
−
1
a_1<a_3<...<a_{2n-1}
a
1
<
a
3
<
...
<
a
2
n
−
1
and
a
2
>
a
4
>
.
.
.
>
a
2
n
.
a_2>a_4>...>a_{2n}.
a
2
>
a
4
>
...
>
a
2
n
.
Prove that
(
a
1
−
a
2
)
2
+
(
a
3
−
a
4
)
2
+
.
.
.
+
(
a
2
n
−
1
−
a
2
n
)
2
>
n
3
(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3
(
a
1
−
a
2
)
2
+
(
a
3
−
a
4
)
2
+
...
+
(
a
2
n
−
1
−
a
2
n
)
2
>
n
3
A4
1
Hide problems
BMO 2014 SL A4
A
4
\boxed{A4}
A
4
Let
m
1
,
m
2
,
m
3
,
n
1
,
n
2
m_1,m_2,m_3,n_1,n_2
m
1
,
m
2
,
m
3
,
n
1
,
n
2
and
n
3
n_3
n
3
be positive real numbers such that
(
m
1
−
n
1
)
(
m
2
−
n
2
)
(
m
3
−
n
3
)
=
m
1
m
2
m
3
−
n
1
n
2
n
3
(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3
(
m
1
−
n
1
)
(
m
2
−
n
2
)
(
m
3
−
n
3
)
=
m
1
m
2
m
3
−
n
1
n
2
n
3
Prove that
(
m
1
+
n
1
)
(
m
2
+
n
2
)
(
m
3
+
n
3
)
≥
8
m
1
m
2
m
3
(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3
(
m
1
+
n
1
)
(
m
2
+
n
2
)
(
m
3
+
n
3
)
≥
8
m
1
m
2
m
3
A3
1
Hide problems
BMO 2014 SL A3
A
3
\boxed{A3}
A
3
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1,a_2,a_3,...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
a
2
=
1
,
a
2
n
+
1
=
2
a
2
n
−
a
n
a_1=a_2=1,a_{2n+1}=2a_{2n}-a_n
a
1
=
a
2
=
1
,
a
2
n
+
1
=
2
a
2
n
−
a
n
and
a
2
n
+
2
=
2
a
2
n
+
1
a_{2n+2}=2a_{2n+1}
a
2
n
+
2
=
2
a
2
n
+
1
for
n
∈
N
.
n\in{N}.
n
∈
N
.
Prove that if
n
>
3
n>3
n
>
3
and
n
−
3
n-3
n
−
3
is divisible by
8
8
8
then
a
n
a_n
a
n
is divisible by
5
5
5
A1
1
Hide problems
Balkan MO SL A1 easy
A1
\boxed{\text{A1}}
A1
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive reals numbers such that
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
.Prove that
2
(
a
2
+
b
2
+
c
2
)
≥
1
9
+
15
a
b
c
2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc
2
(
a
2
+
b
2
+
c
2
)
≥
9
1
+
15
ab
c
G7
1
Hide problems
Balkan MO 2014 shortlist G-7
Let
I
I
I
be the incenter of
△
A
B
C
\triangle ABC
△
A
BC
and let
H
a
H_a
H
a
,
H
b
H_b
H
b
, and
H
c
H_c
H
c
be the orthocenters of
△
B
I
C
\triangle BIC
△
B
I
C
,
△
C
I
A
\triangle CIA
△
C
I
A
, and
△
A
I
B
\triangle AIB
△
A
I
B
, respectively. The lines
H
a
H
b
H_aH_b
H
a
H
b
meets
A
B
AB
A
B
at
X
X
X
and the line
H
a
H
c
H_aH_c
H
a
H
c
meets
A
C
AC
A
C
at
Y
Y
Y
. If the midpoint
T
T
T
of the median
A
M
AM
A
M
of
△
A
B
C
\triangle ABC
△
A
BC
lies on
X
Y
XY
X
Y
, prove that the line
H
a
T
H_aT
H
a
T
is perpendicular to
B
C
BC
BC
G6
1
Hide problems
bisector
In
△
A
B
C
\triangle ABC
△
A
BC
with
A
B
=
A
C
AB=AC
A
B
=
A
C
,
M
M
M
is the midpoint of
B
C
BC
BC
,
H
H
H
is the projection of
M
M
M
onto
A
B
AB
A
B
and
D
D
D
is arbitrary point on the side
A
C
AC
A
C
.Let
E
E
E
be the intersection point of the parallel line through
B
B
B
to
H
D
HD
HD
with the parallel line through
C
C
C
to
A
B
AB
A
B
.Prove that
D
M
DM
D
M
is the bisector of
∠
A
D
E
\angle ADE
∠
A
D
E
.
G5
1
Hide problems
perpendicular segments
Let
A
B
C
D
ABCD
A
BC
D
be a trapezium inscribed in a circle
k
k
k
with diameter
A
B
AB
A
B
. A circle with center
B
B
B
and radius
B
E
BE
BE
,where
E
E
E
is the intersection point of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
meets
k
k
k
at points
K
K
K
and
L
L
L
. If the line ,perpendicular to
B
D
BD
B
D
at
E
E
E
,intersects
C
D
CD
C
D
at
M
M
M
,prove that
K
M
⊥
D
L
KM\perp DL
K
M
⊥
D
L
.
G4
1
Hide problems
Triangle with all coordinates rational
Let
A
0
B
0
C
0
A_0B_0C_0
A
0
B
0
C
0
be a triangle with area equal to
2
\sqrt 2
2
. We consider the excenters
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
then we consider the excenters ,say
A
2
,
B
2
A_2,B_2
A
2
,
B
2
and
C
2
C_2
C
2
,of the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
. By continuing this procedure ,examine if it is possible to arrive to a triangle
A
n
B
n
C
n
A_nB_nC_n
A
n
B
n
C
n
with all coordinates rational.
G3
1
Hide problems
Isosceles triangle
Let
△
A
B
C
\triangle ABC
△
A
BC
be an isosceles.
(
A
B
=
A
C
)
(AB=AC)
(
A
B
=
A
C
)
.Let
D
D
D
and
E
E
E
be two points on the side
B
C
BC
BC
such that
D
∈
B
E
D\in BE
D
∈
BE
,
E
∈
D
C
E\in DC
E
∈
D
C
and
2
∠
D
A
E
=
∠
B
A
C
2\angle DAE = \angle BAC
2∠
D
A
E
=
∠
B
A
C
.Prove that we can construct a triangle
X
Y
Z
XYZ
X
Y
Z
such that
X
Y
=
B
D
XY=BD
X
Y
=
B
D
,
Y
Z
=
D
E
YZ=DE
Y
Z
=
D
E
and
Z
X
=
E
C
ZX=EC
ZX
=
EC
.Find
∠
B
A
C
+
∠
Y
X
Z
\angle BAC + \angle YXZ
∠
B
A
C
+
∠
Y
XZ
.
G2
1
Hide problems
balkan G-2 2014
Triangle
A
B
C
ABC
A
BC
is said to be perpendicular to triangle
D
E
F
DEF
D
EF
if the perpendiculars from
A
A
A
to
E
F
EF
EF
,from
B
B
B
to
F
D
FD
F
D
and from
C
C
C
to
D
E
DE
D
E
are concurrent.Prove that if
A
B
C
ABC
A
BC
is perpendicular to
D
E
F
DEF
D
EF
,then
D
E
F
DEF
D
EF
is perpendicular to
A
B
C
ABC
A
BC
G1
1
Hide problems
Balkan MO 2014 shortlist-G1
Let
A
B
C
ABC
A
BC
be an isosceles triangle, in which
A
B
=
A
C
AB=AC
A
B
=
A
C
, and let
M
M
M
and
N
N
N
be two points on the sides
B
C
BC
BC
and
A
C
AC
A
C
, respectively such that
∠
B
A
M
=
∠
M
N
C
\angle BAM = \angle MNC
∠
B
A
M
=
∠
MNC
. Suppose that the lines
M
N
MN
MN
and
A
B
AB
A
B
intersects at
P
P
P
. Prove that the bisectors of the angles
∠
B
A
M
\angle BAM
∠
B
A
M
and
∠
B
P
M
\angle BPM
∠
BPM
intersects at a point lying on the line
B
C
BC
BC
N6
1
Hide problems
function
Let
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
be a function from the positive integers to the positive integers for which
f
(
1
)
=
1
,
f
(
2
n
)
=
f
(
n
)
f(1)=1,f(2n)=f(n)
f
(
1
)
=
1
,
f
(
2
n
)
=
f
(
n
)
and
f
(
2
n
+
1
)
=
f
(
n
)
+
f
(
n
+
1
)
f(2n+1)=f(n)+f(n+1)
f
(
2
n
+
1
)
=
f
(
n
)
+
f
(
n
+
1
)
for all
n
∈
N
n\in \mathbb{N}
n
∈
N
. Prove that for any natural number
n
n
n
, the number of odd natural numbers
m
m
m
such that
f
(
m
)
=
n
f(m)=n
f
(
m
)
=
n
is equal to the number of positive integers not greater than
n
n
n
having no common prime factors with
n
n
n
.
C2
1
Hide problems
Nonempty subsets and circle !
Let
M
=
{
1
,
2
,
.
.
.
,
2013
}
M=\{1,2,...,2013\}
M
=
{
1
,
2
,
...
,
2013
}
and let
Γ
\Gamma
Γ
be a circle. For every nonempty subset
B
B
B
of the set
M
M
M
, denote by
S
(
B
)
S(B)
S
(
B
)
sum of elements of the set
B
B
B
, and define
S
(
∅
)
=
0
S(\varnothing)=0
S
(
∅
)
=
0
(
∅
\varnothing
∅
is the empty set ). Is it possible to join every subset
B
B
B
of
M
M
M
with some point
A
A
A
on the circle
Γ
\Gamma
Γ
so that following conditions are fulfilled:
1
1
1
. Different subsets are joined with different points;
2
2
2
. All joined points are vertices of a regular polygon;
3
3
3
. If
A
1
,
A
2
,
.
.
.
,
A
k
A_1,A_2,...,A_k
A
1
,
A
2
,
...
,
A
k
are some of the joined points,
k
>
2
k>2
k
>
2
, such that
A
1
A
2
.
.
.
A
k
A_1A_2...A_k
A
1
A
2
...
A
k
is a regular
k
−
g
o
n
k-gon
k
−
g
o
n
, then
2014
2014
2014
divides
S
(
B
1
)
+
S
(
B
2
)
+
.
.
.
+
S
(
B
k
)
S(B_1)+S(B_2)+...+S(B_k)
S
(
B
1
)
+
S
(
B
2
)
+
...
+
S
(
B
k
)
?