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National and Regional Contests
Poland Contests
Poland - Second Round
1984 Poland - Second Round
3
3
Part of
1984 Poland - Second Round
Problems
(1)
(x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) }
Source: Polish MO Recond Round 1984 p3
9/9/2024
The given sequences are
(
x
1
,
x
2
,
…
,
x
n
)
(x_1, x_2, \ldots, x_n)
(
x
1
,
x
2
,
…
,
x
n
)
,
(
y
1
,
y
2
,
…
,
y
n
)
(y_1, y_2, \ldots, y_n)
(
y
1
,
y
2
,
…
,
y
n
)
with positive terms. Prove that there exists a permutation
p
p
p
of the set
{
1
,
2
,
…
,
n
}
\{1, 2, \ldots, n\}
{
1
,
2
,
…
,
n
}
such that for every real
t
t
t
the sequence
(
x
p
(
1
)
+
t
y
p
(
1
)
,
x
p
(
2
)
+
t
y
p
(
2
)
,
…
,
x
p
(
n
)
+
t
y
p
(
n
)
)
(x_{p(1)}+ty_{p(1)}, x_{p(2)}+ty_{p(2)}, \ldots, x_{p(n)}+ty_{p(n) })
(
x
p
(
1
)
+
t
y
p
(
1
)
,
x
p
(
2
)
+
t
y
p
(
2
)
,
…
,
x
p
(
n
)
+
t
y
p
(
n
)
)
has the following property: there is a number
k
k
k
such that
1
≤
k
≤
n
1 \leq k \leq n
1
≤
k
≤
n
and all non-zero terms of the sequence with indices less than
k
k
k
are of the same sign and all non-zero terms of the sequence with indices not less than
k
k
k
are the same sign.
combinatorics
algebra