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Problems
Contests
International Contests
Balkan MO
1985 Balkan MO
1985 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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an international meeting - classic-
There are
1985
1985
1985
participants to an international meeting. In any group of three participants there are at least two who speak the same language. It is known that each participant speaks at most five languages. Prove that there exist at least
200
200
200
participans who speak the same language.
3
1
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set of positive integers of the form 19a+85b
Let
S
S
S
be the set of all positive integers of the form
19
a
+
85
b
19a+85b
19
a
+
85
b
, where
a
,
b
a,b
a
,
b
are arbitrary positive integers. On the real axis, the points of
S
S
S
are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point
A
A
A
on the real axis such that any two points with integer coordinates which are symmetrical with respect to
A
A
A
have necessarily distinct colors.
2
1
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trigonometry
Let
a
,
b
,
c
,
d
∈
[
−
π
2
,
π
2
]
a,b,c,d \in [-\frac{\pi}{2}, \frac{\pi}{2}]
a
,
b
,
c
,
d
∈
[
−
2
π
,
2
π
]
be real numbers such that
sin
a
+
sin
b
+
sin
c
+
sin
d
=
1
\sin{a}+\sin{b}+\sin{c}+\sin{d}=1
sin
a
+
sin
b
+
sin
c
+
sin
d
=
1
and
cos
2
a
+
cos
2
b
+
cos
2
c
+
cos
2
d
≥
10
3
\cos{2a}+\cos{2b}+\cos{2c}+\cos{2d}\geq \frac{10}{3}
cos
2
a
+
cos
2
b
+
cos
2
c
+
cos
2
d
≥
3
10
. Prove that
a
,
b
,
c
,
d
∈
[
0
,
π
6
]
a,b,c,d \in [0, \frac{\pi}{6}]
a
,
b
,
c
,
d
∈
[
0
,
6
π
]
1
1
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perpendicularity if and only if AB=AC
In a given triangle
A
B
C
ABC
A
BC
,
O
O
O
is its circumcenter,
D
D
D
is the midpoint of
A
B
AB
A
B
and
E
E
E
is the centroid of the triangle
A
C
D
ACD
A
C
D
. Show that the lines
C
D
CD
C
D
and
O
E
OE
OE
are perpendicular if and only if
A
B
=
A
C
AB=AC
A
B
=
A
C
.