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International Contests
Balkan MO Shortlist
2019 Balkan MO Shortlist
2019 Balkan MO Shortlist
Part of
Balkan MO Shortlist
Subcontests
(16)
G9
1
Hide problems
4 lines concurrent , starting with semicircle , CD = CH = CZ
Given semicircle
(
c
)
(c)
(
c
)
with diameter
A
B
AB
A
B
and center
O
O
O
. On the
(
c
)
(c)
(
c
)
we take point
C
C
C
such that the tangent at the
C
C
C
intersects the line
A
B
AB
A
B
at the point
E
E
E
. The perpendicular line from
C
C
C
to
A
B
AB
A
B
intersects the diameter
A
B
AB
A
B
at the point
D
D
D
. On the
(
c
)
(c)
(
c
)
we get the points
H
,
Z
H,Z
H
,
Z
such that
C
D
=
C
H
=
C
Z
CD = CH = CZ
C
D
=
C
H
=
CZ
. The line
H
Z
HZ
H
Z
intersects the lines
C
O
,
C
D
,
A
B
CO,CD,AB
CO
,
C
D
,
A
B
at the points
S
,
I
,
K
S, I, K
S
,
I
,
K
respectively and the parallel line from
I
I
I
to the line
A
B
AB
A
B
intersects the lines
C
O
,
C
K
CO,CK
CO
,
C
K
at the points
L
,
M
L,M
L
,
M
respectively. We consider the circumcircle
(
k
)
(k)
(
k
)
of the triangle
L
M
D
LMD
L
M
D
, which intersects again the lines
A
B
,
C
K
AB, CK
A
B
,
C
K
at the points
P
,
U
P, U
P
,
U
respectively. Let
(
e
1
)
,
(
e
2
)
,
(
e
3
)
(e_1), (e_2), (e_3)
(
e
1
)
,
(
e
2
)
,
(
e
3
)
be the tangents of the
(
k
)
(k)
(
k
)
at the points
L
,
M
,
P
L, M, P
L
,
M
,
P
respectively and
R
=
(
e
1
)
∩
(
e
2
)
R = (e_1) \cap (e_2)
R
=
(
e
1
)
∩
(
e
2
)
,
X
=
(
e
2
)
∩
(
e
3
)
X = (e_2) \cap (e_3)
X
=
(
e
2
)
∩
(
e
3
)
,
T
=
(
e
1
)
∩
(
e
3
)
T = (e_1) \cap (e_3)
T
=
(
e
1
)
∩
(
e
3
)
. Prove that if
Q
Q
Q
is the center of
(
k
)
(k)
(
k
)
, then the lines
R
D
,
T
U
,
X
S
RD, TU, XS
R
D
,
T
U
,
XS
pass through the same point, which lies in the line
I
Q
IQ
I
Q
.
G8
1
Hide problems
2 circumcircles and 1 line concurrent, symmetrix wrt sides, orthocenter
Given an acute triangle
A
B
C
ABC
A
BC
,
(
c
)
(c)
(
c
)
its circumcircle with center
O
O
O
and
H
H
H
the orthocenter of the triangle
A
B
C
ABC
A
BC
. The line
A
O
AO
A
O
intersects
(
c
)
(c)
(
c
)
at the point
D
D
D
. Let
D
1
,
D
2
D_1, D_2
D
1
,
D
2
and
H
2
,
H
3
H_2, H_3
H
2
,
H
3
be the symmetrical points of the points
D
D
D
and
H
H
H
with respect to the lines
A
B
,
A
C
AB, AC
A
B
,
A
C
respectively. Let
(
c
1
)
(c_1)
(
c
1
)
be the circumcircle of the triangle
A
D
1
D
2
AD_1D_2
A
D
1
D
2
. Suppose that the line
A
H
AH
A
H
intersects again
(
c
1
)
(c_1)
(
c
1
)
at the point
U
U
U
, the line
H
2
H
3
H_2H_3
H
2
H
3
intersects the segment
D
1
D
2
D_1D_2
D
1
D
2
at the point
K
1
K_1
K
1
and the line
D
H
3
DH_3
D
H
3
intersects the segment
U
D
2
UD_2
U
D
2
at the point
L
1
L_1
L
1
. Prove that one of the intersection points of the circumcircles of the triangles
D
1
K
1
H
2
D_1K_1H_2
D
1
K
1
H
2
and
U
D
L
1
UDL_1
U
D
L
1
lies on the line
K
1
L
1
K_1L_1
K
1
L
1
.
G7
1
Hide problems
concurrency wanted, reflections of feet of altitudes, orthocenter
Let
A
D
,
B
E
AD, BE
A
D
,
BE
, and
C
F
CF
CF
denote the altitudes of triangle
△
A
B
C
\vartriangle ABC
△
A
BC
. Points
E
′
E'
E
′
and
F
′
F'
F
′
are the reflections of
E
E
E
and
F
F
F
over
A
D
AD
A
D
, respectively. The lines
B
F
′
BF'
B
F
′
and
C
E
′
CE'
C
E
′
intersect at
X
X
X
, while the lines
B
E
′
BE'
B
E
′
and
C
F
′
CF'
C
F
′
intersect at the point
Y
Y
Y
. Prove that if
H
H
H
is the orthocenter of
△
A
B
C
\vartriangle ABC
△
A
BC
, then the lines
A
X
,
Y
H
AX, YH
A
X
,
Y
H
, and
B
C
BC
BC
are concurrent.
G5
1
Hide problems
TB = TQ wanted, 5 circumcircles related
Let
A
B
C
ABC
A
BC
(
B
C
>
A
C
BC > AC
BC
>
A
C
) be an acute triangle with circumcircle
k
k
k
centered at
O
O
O
. The tangent to
k
k
k
at
C
C
C
intersects the line
A
B
AB
A
B
at the point
D
D
D
. The circumcircles of triangles
B
C
D
,
O
C
D
BCD, OCD
BC
D
,
OC
D
and
A
O
B
AOB
A
OB
intersect the ray
C
A
CA
C
A
(beyond
A
A
A
) at the points
Q
,
P
Q, P
Q
,
P
and
K
K
K
, respectively, such that
P
∈
(
A
K
)
P \in (AK)
P
∈
(
A
K
)
and
K
∈
(
P
Q
)
K \in (PQ)
K
∈
(
PQ
)
. The line
P
D
PD
P
D
intersects the circumcircle of triangle
B
K
Q
BKQ
B
K
Q
at the point
T
T
T
, so that
P
P
P
and
T
T
T
are in different halfplanes with respect to
B
Q
BQ
BQ
. Prove that
T
B
=
T
Q
TB = TQ
TB
=
TQ
.
C1
1
Hide problems
Moldovan traditional dance
100 couples are invited to a traditional Modolvan dance. The
200
200
200
people stand in a line, and then in a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
t
e
p
<
/
s
p
a
n
>
<span class='latex-italic'>step</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
t
e
p
<
/
s
p
an
>
, (not necessarily adjacent) many swap positions. Find the least
C
C
C
such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most
C
C
C
steps.
C4
1
Hide problems
Vlad wants to drive
A town-planner has built an isolated city whose road network consists of
2
N
2N
2
N
roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise. Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of
N
N
N
is this possible?
G3
1
Hide problems
Prove tangent meet at circumcircle (ABC)
Let
A
B
C
ABC
A
BC
be a scalene and acute triangle with circumcenter
O
O
O
. Let
ω
\omega
ω
be the circle with center
A
A
A
, tangent to
B
C
BC
BC
at
D
D
D
. Suppose there are two points
F
F
F
and
G
G
G
on
ω
\omega
ω
such that
F
G
⊥
A
O
FG \perp AO
FG
⊥
A
O
,
∠
B
F
D
=
∠
D
G
C
\angle BFD = \angle DGC
∠
BF
D
=
∠
D
GC
and the couples of points
(
B
,
F
)
(B,F)
(
B
,
F
)
and
(
C
,
G
)
(C,G)
(
C
,
G
)
are in different halfplanes with respect to the line
A
D
AD
A
D
. Show that the tangents to
ω
\omega
ω
at
F
F
F
and
G
G
G
meet on the circumcircle of
A
B
C
ABC
A
BC
.
G2
1
Hide problems
midpoint of segment in a triangle 75-60-45
Let be a triangle
△
A
B
C
\triangle ABC
△
A
BC
with
m
(
∠
A
B
C
)
=
7
5
∘
m(\angle ABC) = 75^{\circ}
m
(
∠
A
BC
)
=
7
5
∘
and
m
(
∠
A
C
B
)
=
4
5
∘
m(\angle ACB) = 45^{\circ}
m
(
∠
A
CB
)
=
4
5
∘
. The angle bisector of
∠
C
A
B
\angle CAB
∠
C
A
B
intersects
C
B
CB
CB
at point
D
D
D
. We consider the point
E
∈
(
A
B
)
E \in (AB)
E
∈
(
A
B
)
, such that
D
E
=
D
C
DE = DC
D
E
=
D
C
. Let
P
P
P
be the intersection of lines
A
D
AD
A
D
and
C
E
CE
CE
. Prove that
P
P
P
is the midpoint of segment
A
D
AD
A
D
.
G1
1
Hide problems
square geometry bisect $\angle ESB$
Let
A
B
C
D
ABCD
A
BC
D
be a square of center
O
O
O
and let
M
M
M
be the symmetric of the point
B
B
B
with respect to point
A
A
A
. Let
E
E
E
be the intersection of
C
M
CM
CM
and
B
D
BD
B
D
, and let
S
S
S
be the intersection of
M
O
MO
MO
and
A
E
AE
A
E
. Show that
S
O
SO
SO
is the angle bisector of
∠
E
S
B
\angle ESB
∠
ESB
.
A2
1
Hide problems
f(xy) = yf(x) + x + f(f(y) - f(x))
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
y
)
=
y
f
(
x
)
+
x
+
f
(
f
(
y
)
−
f
(
x
)
)
f(xy) = yf(x) + x + f(f(y) - f(x))
f
(
x
y
)
=
y
f
(
x
)
+
x
+
f
(
f
(
y
)
−
f
(
x
))
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.
A4
1
Hide problems
another inequality wrapped in sigma and power of -1
Let
a
i
j
,
i
=
1
,
2
,
…
,
m
a_{ij}, i = 1, 2, \dots, m
a
ij
,
i
=
1
,
2
,
…
,
m
and
j
=
1
,
2
,
…
,
n
j = 1, 2, \dots, n
j
=
1
,
2
,
…
,
n
be positive real numbers. Prove that
∑
i
=
1
m
(
∑
j
=
1
n
1
a
i
j
)
−
1
≤
(
∑
j
=
1
n
(
∑
i
=
1
m
a
i
j
)
−
1
)
−
1
\sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1}
i
=
1
∑
m
(
j
=
1
∑
n
a
ij
1
)
−
1
≤
j
=
1
∑
n
(
i
=
1
∑
m
a
ij
)
−
1
−
1
G4
1
Hide problems
classic geometry
Given an acute triangle
A
B
C
ABC
A
BC
, let
M
M
M
be the midpoint of
B
C
BC
BC
and
H
H
H
the orthocentre. Let
Γ
\Gamma
Γ
be the circle with diameter
H
M
HM
H
M
, and let
X
,
Y
X,Y
X
,
Y
be distinct points on
Γ
\Gamma
Γ
such that
A
X
,
A
Y
AX,AY
A
X
,
A
Y
are tangent to
Γ
\Gamma
Γ
. Prove that
B
X
Y
C
BXYC
BX
Y
C
is cyclic.
A5
1
Hide problems
(ab)^2 + (bc)^2 + (ca)^2
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers, such that
(
a
b
)
2
+
(
b
c
)
2
+
(
c
a
)
2
=
3
(ab)^2 + (bc)^2 + (ca)^2 = 3
(
ab
)
2
+
(
b
c
)
2
+
(
c
a
)
2
=
3
. Prove that
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
(
c
2
−
c
+
1
)
≥
1.
(a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1.
(
a
2
−
a
+
1
)
(
b
2
−
b
+
1
)
(
c
2
−
c
+
1
)
≥
1.
Proposed by Florin Stanescu (wer), România
N2
1
Hide problems
gcd of set $S$
Let
S
⊂
{
1
,
…
,
n
}
S \subset \{ 1, \dots, n \}
S
⊂
{
1
,
…
,
n
}
be a nonempty set, where
n
n
n
is a positive integer. We denote by
s
s
s
the greatest common divisor of the elements of the set
S
S
S
. We assume that
s
≠
1
s \not= 1
s
=
1
and let
d
d
d
be its smallest divisor greater than
1
1
1
. Let
T
⊂
{
1
,
…
,
n
}
T \subset \{ 1, \dots, n \}
T
⊂
{
1
,
…
,
n
}
be a set such that
S
⊂
T
S \subset T
S
⊂
T
and
∣
T
∣
≥
1
+
[
n
d
]
|T| \ge 1 + \left[ \frac{n}{d} \right]
∣
T
∣
≥
1
+
[
d
n
]
. Prove that the greatest common divisor of the elements in
T
T
T
is
1
1
1
. ———-- [Second Version] Let
n
(
n
≥
1
)
n(n \ge 1)
n
(
n
≥
1
)
be a positive integer and
U
=
{
1
,
…
,
n
}
U = \{ 1, \dots, n \}
U
=
{
1
,
…
,
n
}
. Let
S
S
S
be a nonempty subset of
U
U
U
and let
d
(
d
≠
1
)
d (d \not= 1)
d
(
d
=
1
)
be the smallest common divisor of all elements of the set
S
S
S
. Find the smallest positive integer
k
k
k
such that for any subset
T
T
T
of
U
U
U
, consisting of
k
k
k
elements, with
S
⊂
T
S \subset T
S
⊂
T
, the greatest common divisor of all elements of
T
T
T
is equal to
1
1
1
.
A1
1
Hide problems
Sequence a_(n+1)=a_n
Let
a
0
a_0
a
0
be an arbitrary positive integer. Consider the infinite sequence
(
a
n
)
n
≥
1
(a_n)_{n\geq 1}
(
a
n
)
n
≥
1
, defined inductively as follows: given
a
0
,
a
1
,
.
.
.
,
a
n
−
1
a_0, a_1, ..., a_{n-1}
a
0
,
a
1
,
...
,
a
n
−
1
define the term
a
n
a_n
a
n
as the smallest positive integer such that
a
0
+
a
1
+
.
.
.
+
a
n
a_0+a_1+...+a_n
a
0
+
a
1
+
...
+
a
n
is divisible by
n
n
n
. Prove that there exist a positive integer a positive integer
M
M
M
such that
a
n
+
1
=
a
n
a_{n+1}=a_n
a
n
+
1
=
a
n
for all
n
≥
M
n\geq M
n
≥
M
.
C2
1
Hide problems
5x5 arrays defined in function of 2x2 subarrays
Determine the largest natural number
N
N
N
having the following property: every
5
×
5
5\times 5
5
×
5
array consisting of pairwise distinct natural numbers from
1
1
1
to
25
25
25
contains a
2
×
2
2\times 2
2
×
2
subarray of numbers whose sum is, at least,
N
.
N.
N
.
Demetres Christofides and Silouan Brazitikos