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Contests
International Contests
Balkan MO
1999 Balkan MO
1999 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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Non-decreasing unbounded sequence of non-negative integers
Let
{
a
n
}
n
≥
0
\{a_n\}_{n\geq 0}
{
a
n
}
n
≥
0
be a non-decreasing, unbounded sequence of non-negative integers with
a
0
=
0
a_0=0
a
0
=
0
. Let the number of members of the sequence not exceeding
n
n
n
be
b
n
b_n
b
n
. Prove that
(
a
0
+
a
1
+
⋯
+
a
m
)
(
b
0
+
b
1
+
⋯
+
b
n
)
≥
(
m
+
1
)
(
n
+
1
)
.
(a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1).
(
a
0
+
a
1
+
⋯
+
a
m
)
(
b
0
+
b
1
+
⋯
+
b
n
)
≥
(
m
+
1
)
(
n
+
1
)
.
1
1
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Let O be the circumcenter of the triangle ABC
Let
O
O
O
be the circumcenter of the triangle
A
B
C
ABC
A
BC
. The segment
X
Y
XY
X
Y
is the diameter of the circumcircle perpendicular to
B
C
BC
BC
and it meets
B
C
BC
BC
at
M
M
M
. The point
X
X
X
is closer to
M
M
M
than
Y
Y
Y
and
Z
Z
Z
is the point on
M
Y
MY
M
Y
such that
M
Z
=
M
X
MZ = MX
MZ
=
MX
. The point
W
W
W
is the midpoint of
A
Z
AZ
A
Z
. a) Show that
W
W
W
lies on the circle through the midpoints of the sides of
A
B
C
ABC
A
BC
; b) Show that
M
W
MW
M
W
is perpendicular to
A
Y
AY
A
Y
.
3
1
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ABC is an acute-angled triangle with area 1
Let
A
B
C
ABC
A
BC
be an acute-angled triangle of area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid
G
G
G
to
A
B
AB
A
B
,
B
C
BC
BC
,
C
A
CA
C
A
has area between
4
27
\frac 4{27}
27
4
and
1
4
\frac 14
4
1
.
2
1
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p is an odd prime congruent to 2 mod 3
Let
p
p
p
be an odd prime congruent to 2 modulo 3. Prove that at most
p
−
1
p-1
p
−
1
members of the set
{
m
2
−
n
3
−
1
∣
0
<
m
,
n
<
p
}
\{m^2 - n^3 - 1 \mid 0 < m,\ n < p\}
{
m
2
−
n
3
−
1
∣
0
<
m
,
n
<
p
}
are divisible by
p
p
p
.