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Contests
International Contests
Balkan MO
1991 Balkan MO
1991 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
Hide problems
prove there isn't any bijection
Prove that there is no bijective function
f
:
{
1
,
2
,
3
,
…
}
→
{
0
,
1
,
2
,
3
,
…
}
f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}
f
:
{
1
,
2
,
3
,
…
}
→
{
0
,
1
,
2
,
3
,
…
}
such that
f
(
m
n
)
=
f
(
m
)
+
f
(
n
)
+
3
f
(
m
)
f
(
n
)
f(mn)=f(m)+f(n)+3f(m)f(n)
f
(
mn
)
=
f
(
m
)
+
f
(
n
)
+
3
f
(
m
)
f
(
n
)
.
3
1
Hide problems
a bulgarian beauty, regular hexagon inscribed in a polygon
A regular hexagon of area
H
H
H
is inscribed in a convex polygon of area
P
P
P
. Show that
P
≤
3
2
H
P \leq \frac{3}{2}H
P
≤
2
3
H
. When does the equality occur?
2
1
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a geometrical problem with a number teory flavour
Show that there are infinitely many noncongruent triangles which satisfy the following conditions: i) the side lengths are relatively prime integers; ii)the area is an integer number; iii)the altitudes' lengths are not integer numbers.
1
1
Hide problems
find the size of an angle, a nice problem proposed by Greece
Let
A
B
C
ABC
A
BC
be an acute triangle inscribed in a circle centered at
O
O
O
. Let
M
M
M
be a point on the small arc
A
B
AB
A
B
of the triangle's circumcircle. The perpendicular dropped from
M
M
M
on the ray
O
A
OA
O
A
intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at the points
K
K
K
and
L
L
L
, respectively. Similarly, the perpendicular dropped from
M
M
M
on the ray
O
B
OB
OB
intersects the sides
A
B
AB
A
B
and
B
C
BC
BC
at
N
N
N
and
P
P
P
, respectively. Assume that
K
L
=
M
N
KL=MN
K
L
=
MN
. Find the size of the angle
∠
M
L
P
\angle{MLP}
∠
M
L
P
in terms of the angles of the triangle
A
B
C
ABC
A
BC
.