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Contests
International Contests
Balkan MO Shortlist
2012 Balkan MO Shortlist
2012 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(13)
G2
1
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line passes through orthocenter iff AB=AC or <BCA=120^o, circumcenter related
Let
A
B
C
ABC
A
BC
be a triangle, and let
ℓ
\ell
ℓ
be the line passing through the circumcenter of
A
B
C
ABC
A
BC
and parallel to the bisector of the angle
∠
A
\angle A
∠
A
. Prove that the line
ℓ
\ell
ℓ
passes through the orthocenter of
A
B
C
ABC
A
BC
if and only if
A
B
=
A
C
AB = AC
A
B
=
A
C
or
∠
B
A
C
=
12
0
o
\angle BAC = 120^o
∠
B
A
C
=
12
0
o
A3
1
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Maximum possible number of distinct real roots
Determine the maximum possible number of distinct real roots of a polynomial
P
(
x
)
P(x)
P
(
x
)
of degree
2012
2012
2012
with real coefficients satisfying the condition \begin{align*} P(a)^3 + P(b)^3 + P(c)^3 \geq 3 P(a) P(b) P(c) \end{align*} for all real numbers
a
,
b
,
c
∈
R
a,b,c \in \mathbb{R}
a
,
b
,
c
∈
R
with
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
N2
1
Hide problems
c^2_r=c_kc_m, c_n=a_n+b_n,a_n =9a_{n-1}-2b_{n-1}, b_n=2a_{n-1}+4b_{n-1},
Let the sequences
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
and
(
b
n
)
n
=
1
∞
(b_n)_{n=1}^{\infty}
(
b
n
)
n
=
1
∞
satisfy
a
0
=
b
0
=
1
,
a
n
=
9
a
n
−
1
−
2
b
n
−
1
a_0 = b_0 = 1, a_n = 9a_{n-1} -2b_{n-1}
a
0
=
b
0
=
1
,
a
n
=
9
a
n
−
1
−
2
b
n
−
1
and
b
n
=
2
a
n
−
1
+
4
b
n
−
1
b_n = 2a_{n-1} + 4b_{n-1}
b
n
=
2
a
n
−
1
+
4
b
n
−
1
for all positive integers
n
n
n
. Let
c
n
=
a
n
+
b
n
c_n = a_n + b_n
c
n
=
a
n
+
b
n
for all positive integers
n
n
n
. Prove that there do not exist positive integers
k
,
r
,
m
k, r, m
k
,
r
,
m
such that
c
r
2
=
c
k
c
m
c^2_r = c_kc_m
c
r
2
=
c
k
c
m
.
N1
1
Hide problems
a_{n+1} = a_n +\tau (n), consective terms are perfect squares
A sequence
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
of positive integers satisfies the condition
a
n
+
1
=
a
n
+
τ
(
n
)
a_{n+1} = a_n +\tau (n)
a
n
+
1
=
a
n
+
τ
(
n
)
for all positive integers
n
n
n
where
τ
(
n
)
\tau (n)
τ
(
n
)
is the number of positive integer divisors of
n
n
n
. Determine whether two consecutive terms of this sequence can be perfect squares.
G4
1
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K is circumcenter of triangle SVG, incenters, circumcenters, cyclic related
Let
M
M
M
be the point of intersection of the diagonals of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
. Let
I
1
I_1
I
1
and
I
2
I_2
I
2
are the incenters of triangles
A
M
D
AMD
A
M
D
and
B
M
C
BMC
BMC
, respectively, and let
L
L
L
be the point of intersection of the lines
D
I
1
DI_1
D
I
1
and
C
I
2
CI_2
C
I
2
. The foot of the perpendicular from the midpoint
T
T
T
of
I
1
I
2
I_1I_2
I
1
I
2
to
C
L
CL
C
L
is
N
N
N
, and
F
F
F
is the midpoint of
T
N
TN
TN
. Let
G
G
G
and
J
J
J
be the points of intersection of the line
L
F
LF
L
F
with
I
1
N
I_1N
I
1
N
and
I
1
I
2
I_1I_2
I
1
I
2
, respectively. Let
O
1
O_1
O
1
be the circumcenter of triangle
L
I
1
J
LI_1J
L
I
1
J
, and let
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
be the circles with diameters
O
1
L
O_1L
O
1
L
and
O
1
J
O_1J
O
1
J
, respectively. Let
V
V
V
and
S
S
S
be the second points of intersection of
I
1
O
1
I_1O_1
I
1
O
1
with
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
, respectively. If
K
K
K
is point where the circles
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
meet again, prove that
K
K
K
is the circumcenter of the triangle
S
V
G
SVG
S
V
G
.
G6
1
Hide problems
perpendicularity wanted, <PAC = <QAB, <PBC = <QBA given
Let
P
P
P
and
Q
Q
Q
be points inside a triangle
A
B
C
ABC
A
BC
such that
∠
P
A
C
=
∠
Q
A
B
\angle PAC = \angle QAB
∠
P
A
C
=
∠
Q
A
B
and
∠
P
B
C
=
∠
Q
B
A
\angle PBC = \angle QBA
∠
PBC
=
∠
QB
A
. Let
D
D
D
and
E
E
E
be the feet of the perpendiculars from
P
P
P
to the lines
B
C
BC
BC
and
A
C
AC
A
C
, and
F
F
F
be the foot of perpendicular from
Q
Q
Q
to the line
A
B
AB
A
B
. Let
M
M
M
be intersection of the lines
D
E
DE
D
E
and
A
B
AB
A
B
. Prove that
M
P
⊥
C
F
MP \perp CF
MP
⊥
CF
G7
1
Hide problems
rectangle wanted, starting with a cyclic ABCD and angle bisectors
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral. The lines
A
D
AD
A
D
and
B
C
BC
BC
meet at X, and the lines
A
B
AB
A
B
and
C
D
CD
C
D
meet at
Y
Y
Y
. The line joining the midpoints
M
M
M
and
N
N
N
of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
, respectively, meets the internal bisector of angle
A
X
B
AXB
A
XB
at
P
P
P
and the external bisector of angle
B
Y
C
BYC
B
Y
C
at
Q
Q
Q
. Prove that
P
X
Q
Y
PXQY
PXQ
Y
is a rectangle
G3
1
Hide problems
congruent triangles, 3 circumcircles related
Let
A
B
C
ABC
A
BC
be a triangle with circumcircle
c
c
c
and circumcenter
O
O
O
, and let
D
D
D
be a point on the side
B
C
BC
BC
different from the vertices and the midpoint of
B
C
BC
BC
. Let
K
K
K
be the point where the circumcircle
c
1
c_1
c
1
of the triangle
B
O
D
BOD
BO
D
intersects
c
c
c
for the second time and let
Z
Z
Z
be the point where
c
1
c_1
c
1
meets the line
A
B
AB
A
B
. Let
M
M
M
be the point where the circumcircle
c
2
c_2
c
2
of the triangle
C
O
D
COD
CO
D
intersects
c
c
c
for the second time and let
E
E
E
be the point where
c
2
c_2
c
2
meets the line
A
C
AC
A
C
. Finally let
N
N
N
be the point where the circumcircle
c
3
c_3
c
3
of the triangle
A
E
Z
AEZ
A
EZ
meets
c
c
c
again. Prove that the triangles
A
B
C
ABC
A
BC
and
N
K
M
NKM
N
K
M
are congruent.
A4
1
Hide problems
min and max of f (P) =\frac{PA + PB}{PC + PD}, where ABCD is a square
Let
A
B
C
D
ABCD
A
BC
D
be a square of the plane
P
P
P
. Define the minimum and the maximum the value of the function
f
:
P
→
R
f: P \to R
f
:
P
→
R
is given by
f
(
P
)
=
P
A
+
P
B
P
C
+
P
D
f (P) =\frac{PA + PB}{PC + PD}
f
(
P
)
=
PC
+
P
D
P
A
+
PB
G5
1
Hide problems
İncircle
G5
\boxed{\text{G5}}
G5
The incircle of a triangle
A
B
C
ABC
A
BC
touches its sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at the points
A
1
A_1
A
1
,
B
1
B_1
B
1
,
C
1
C_1
C
1
.Let the projections of the orthocenter
H
1
H_1
H
1
of the triangle
A
1
B
1
C
1
A_{1}B_{1}C_{1}
A
1
B
1
C
1
to the lines
A
A
1
AA_1
A
A
1
and
B
C
BC
BC
be
P
P
P
and
Q
Q
Q
,respectively. Show that
P
Q
PQ
PQ
bisects the line segment
B
1
C
1
B_{1}C_{1}
B
1
C
1
A2
1
Hide problems
A nice inequality with a+b+c=sqrt2
Let
a
,
b
,
c
≥
0
a,b,c\ge 0
a
,
b
,
c
≥
0
and
a
+
b
+
c
=
2
a+b+c=\sqrt2
a
+
b
+
c
=
2
. Show that
1
1
+
a
2
+
1
1
+
b
2
+
1
1
+
c
2
≥
2
+
1
3
\frac1{\sqrt{1+a^2}}+\frac1{\sqrt{1+b^2}}+\frac1{\sqrt{1+c^2}} \ge 2+\frac1{\sqrt3}
1
+
a
2
1
+
1
+
b
2
1
+
1
+
c
2
1
≥
2
+
3
1
In general if
a
1
,
a
2
,
⋯
,
a
n
≥
0
a_1, a_2, \cdots , a_n \ge 0
a
1
,
a
2
,
⋯
,
a
n
≥
0
and
∑
i
=
1
n
a
i
=
2
\sum_{i=1}^n a_i=\sqrt2
∑
i
=
1
n
a
i
=
2
we have
∑
i
=
1
n
1
1
+
a
i
2
≥
(
n
−
1
)
+
1
3
\sum_{i=1}^n \frac1{\sqrt{1+a_i^2}} \ge (n-1)+\frac1{\sqrt3}
i
=
1
∑
n
1
+
a
i
2
1
≥
(
n
−
1
)
+
3
1
A5
1
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Inequality involving some functions
Let
f
,
g
:
Z
→
[
0
,
∞
)
f, g:\mathbb{Z}\rightarrow [0,\infty )
f
,
g
:
Z
→
[
0
,
∞
)
be two functions such that
f
(
n
)
=
g
(
n
)
=
0
f(n)=g(n)=0
f
(
n
)
=
g
(
n
)
=
0
with the exception of finitely many integers
n
n
n
. Define
h
:
Z
→
[
0
,
∞
)
h:\mathbb{Z}\rightarrow [0,\infty )
h
:
Z
→
[
0
,
∞
)
by
h
(
n
)
=
max
{
f
(
n
−
k
)
g
(
k
)
:
k
∈
Z
}
.
h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.
h
(
n
)
=
max
{
f
(
n
−
k
)
g
(
k
)
:
k
∈
Z
}
.
Let
p
p
p
and
q
q
q
be two positive reals such that
1
/
p
+
1
/
q
=
1
1/p+1/q=1
1/
p
+
1/
q
=
1
. Prove that
∑
n
∈
Z
h
(
n
)
≥
(
∑
n
∈
Z
f
(
n
)
p
)
1
/
p
(
∑
n
∈
Z
g
(
n
)
q
)
1
/
q
.
\sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.
n
∈
Z
∑
h
(
n
)
≥
(
n
∈
Z
∑
f
(
n
)
p
)
1/
p
(
n
∈
Z
∑
g
(
n
)
q
)
1/
q
.
A6
1
Hide problems
Maximum value of cyclic expression
Let
k
k
k
be a positive integer. Find the maximum value of
a
3
k
−
1
b
+
b
3
k
−
1
c
+
c
3
k
−
1
a
+
k
2
a
k
b
k
c
k
,
a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,
a
3
k
−
1
b
+
b
3
k
−
1
c
+
c
3
k
−
1
a
+
k
2
a
k
b
k
c
k
,
where
a
a
a
,
b
b
b
,
c
c
c
are non-negative reals such that
a
+
b
+
c
=
3
k
a+b+c=3k
a
+
b
+
c
=
3
k
.