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China Team Selection Test
2018 China Team Selection Test
1
Existence of reals satisfying cyclic relation
Existence of reals satisfying cyclic relation
Source: 2018 China TST Day 1 Q1
January 2, 2018
algebra
Problem Statement
Let
p
,
q
p,q
p
,
q
be positive reals with sum 1. Show that for any
n
n
n
-tuple of reals
(
y
1
,
y
2
,
.
.
.
,
y
n
)
(y_1,y_2,...,y_n)
(
y
1
,
y
2
,
...
,
y
n
)
, there exists an
n
n
n
-tuple of reals
(
x
1
,
x
2
,
.
.
.
,
x
n
)
(x_1,x_2,...,x_n)
(
x
1
,
x
2
,
...
,
x
n
)
satisfying
p
⋅
max
{
x
i
,
x
i
+
1
}
+
q
⋅
min
{
x
i
,
x
i
+
1
}
=
y
i
p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i
p
⋅
max
{
x
i
,
x
i
+
1
}
+
q
⋅
min
{
x
i
,
x
i
+
1
}
=
y
i
for all
i
=
1
,
2
,
.
.
.
,
2017
i=1,2,...,2017
i
=
1
,
2
,
...
,
2017
, where
x
2018
=
x
1
x_{2018}=x_1
x
2018
=
x
1
.
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