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Contests
International Contests
Junior Balkan MO
2002 Junior Balkan MO
2002 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
3
1
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Find all positive integers with exactly 16 positive divisors
Find all positive integers which have exactly 16 positive divisors
1
=
d
1
<
d
2
<
…
<
d
16
=
n
1 = d_1 < d_2 < \ldots < d_{16} =n
1
=
d
1
<
d
2
<
…
<
d
16
=
n
such that the divisor
d
k
d_k
d
k
, where
k
=
d
5
k = d_5
k
=
d
5
, equals
(
d
2
+
d
4
)
d
6
(d_2 + d_4) d_6
(
d
2
+
d
4
)
d
6
.
1
1
Hide problems
Isosceles triangle and point P on the circumcircle
The triangle
A
B
C
ABC
A
BC
has
C
A
=
C
B
CA = CB
C
A
=
CB
.
P
P
P
is a point on the circumcircle between
A
A
A
and
B
B
B
(and on the opposite side of the line
A
B
AB
A
B
to
C
C
C
).
D
D
D
is the foot of the perpendicular from
C
C
C
to
P
B
PB
PB
. Show that
P
A
+
P
B
=
2
⋅
P
D
PA + PB = 2 \cdot PD
P
A
+
PB
=
2
⋅
P
D
.
2
1
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the centers of the circles are on opposite sides of AB
Two circles with centers
O
1
O_{1}
O
1
and
O
2
O_{2}
O
2
meet at two points
A
A
A
and
B
B
B
such that the centers of the circles are on opposite sides of the line
A
B
AB
A
B
. The lines
B
O
1
BO_{1}
B
O
1
and
B
O
2
BO_{2}
B
O
2
meet their respective circles again at
B
1
B_{1}
B
1
and
B
2
B_{2}
B
2
. Let
M
M
M
be the midpoint of
B
1
B
2
B_{1}B_{2}
B
1
B
2
. Let
M
1
M_{1}
M
1
,
M
2
M_{2}
M
2
be points on the circles of centers
O
1
O_{1}
O
1
and
O
2
O_{2}
O
2
respectively, such that
∠
A
O
1
M
1
=
∠
A
O
2
M
2
\angle AO_{1}M_{1}= \angle AO_{2}M_{2}
∠
A
O
1
M
1
=
∠
A
O
2
M
2
, and
B
1
B_{1}
B
1
lies on the minor arc
A
M
1
AM_{1}
A
M
1
while
B
B
B
lies on the minor arc
A
M
2
AM_{2}
A
M
2
. Show that
∠
M
M
1
B
=
∠
M
M
2
B
\angle MM_{1}B = \angle MM_{2}B
∠
M
M
1
B
=
∠
M
M
2
B
. Ciprus
4
1
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nice inequality by panaitopol
Prove that for all positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
the following inequality takes place
1
b
(
a
+
b
)
+
1
c
(
b
+
c
)
+
1
a
(
c
+
a
)
≥
27
2
(
a
+
b
+
c
)
2
.
\frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} .
b
(
a
+
b
)
1
+
c
(
b
+
c
)
1
+
a
(
c
+
a
)
1
≥
2
(
a
+
b
+
c
)
2
27
.
Laurentiu Panaitopol, Romania