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Balkan MO
1994 Balkan MO
1994 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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Acquaintances
Find the smallest number
n
≥
5
n \geq 5
n
≥
5
for which there can exist a set of
n
n
n
people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.Bulgaria
3
1
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Determine the largest possible value of sum involving permut
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be a permutation of the numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
, with
n
≥
2
n\geq 2
n
≥
2
. Determine the largest possible value of the sum
S
(
n
)
=
∣
a
2
−
a
1
∣
+
∣
a
3
−
a
2
∣
+
⋯
+
∣
a
n
−
a
n
−
1
∣
.
S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| .
S
(
n
)
=
∣
a
2
−
a
1
∣
+
∣
a
3
−
a
2
∣
+
⋯
+
∣
a
n
−
a
n
−
1
∣.
Romania
2
1
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Prove that the polynomial has at most one zero
Let
n
n
n
be an integer. Prove that the polynomial
f
(
x
)
f(x)
f
(
x
)
has at most one zero, where
f
(
x
)
=
x
4
−
1994
x
3
+
(
1993
+
n
)
x
2
−
11
x
+
n
.
f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n .
f
(
x
)
=
x
4
−
1994
x
3
+
(
1993
+
n
)
x
2
−
11
x
+
n
.
Greece
1
1
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An acute angle and a point inside it are given
An acute angle
X
A
Y
XAY
X
A
Y
and a point
P
P
P
inside the angle are given. Construct (using a ruler and a compass) a line that passes through
P
P
P
and intersects the rays
A
X
AX
A
X
and
A
Y
AY
A
Y
at
B
B
B
and
C
C
C
such that the area of the triangle
A
B
C
ABC
A
BC
equals
A
P
2
AP^2
A
P
2
. Greece