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International Contests
Hungary-Israel Binational
1994 Hungary-Israel Binational
1994 Hungary-Israel Binational
Part of
Hungary-Israel Binational
Subcontests
(4)
4
1
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Hungary-Israel Binational 1994_4
An n\minus{}m society is a group of
n
n
n
girls and
m
m
m
boys. Prove that there exists numbers
n
0
n_0
n
0
and
m
0
m_0
m
0
such that every n_0\minus{}m_0 society contains a subgroup of five boys and five girls with the following property: either all of the boys know all of the girls or none of the boys knows none of the girls.
3
1
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Hungary-Israel Binational 1994_3
Three given circles have the same radius and pass through a common point
P
P
P
. Their other points of pairwise intersections are
A
A
A
,
B
B
B
,
C
C
C
. We define triangle
A
′
B
′
C
′
A'B'C'
A
′
B
′
C
′
, each of whose sides is tangent to two of the three circles. The three circles are contained in
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
. Prove that the area of
△
A
′
B
′
C
′
\triangle A'B'C'
△
A
′
B
′
C
′
is at least nine times the area of
△
A
B
C
\triangle ABC
△
A
BC
2
1
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Hungary-Israel Binational 1994_2
Let
a
1
a_1
a
1
,
…
\ldots
…
,
a
k
a_k
a
k
, a_{k\plus{}1},
…
\ldots
…
,
a
n
a_n
a
n
be
n
n
n
positive numbers (
k
<
n
k<n
k
<
n
). Suppose that the values of a_{k\plus{}1}, a_{k\plus{}2},
…
\ldots
…
,
a
n
a_n
a
n
are fixed. Choose the values of
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\ldots
…
,
a
k
a_k
a
k
that minimize the sum
∑
i
,
j
,
i
≠
j
a
i
a
j
\sum_{i, j, i\neq j}\frac{a_i}{a_j}
∑
i
,
j
,
i
=
j
a
j
a
i
1
1
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Hungary-Israel Binational 1994_1
Let
m
m
m
and
n
n
n
be two distinct positive integers. Prove that there exists a real number
x
x
x
such that
1
3
≤
{
x
n
}
≤
2
3
\frac {1}{3}\le\{xn\}\le\frac {2}{3}
3
1
≤
{
x
n
}
≤
3
2
and
1
3
≤
{
x
m
}
≤
2
3
\frac {1}{3}\le\{xm\}\le\frac {2}{3}
3
1
≤
{
x
m
}
≤
3
2
. Here, for any real number
y
y
y
,
{
y
}
\{y\}
{
y
}
denotes the fractional part of
y
y
y
. For example \{3.1415\} \equal{} 0.1415.