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1992 Tournament Of Towns
(351) 3
(351) 3
Part of
1992 Tournament Of Towns
Problems
(1)
TOT 351 1992 Autumn O S3 total length of intervals increasing = ... decreasing
Source:
6/10/2024
We are given a finite number of functions of the form
y
=
c
2
−
∣
x
−
d
∣
y = c2^{-|x-d|}
y
=
c
2
−
∣
x
−
d
∣
. In each case
c
c
c
and
d
d
d
are parameters with
c
>
0
c > 0
c
>
0
. The function
f
(
x
)
f(x)
f
(
x
)
is defined on the interval
[
a
,
b
]
[a, b]
[
a
,
b
]
as follows: For each
x
x
x
in
[
a
,
b
]
[a, b]
[
a
,
b
]
,
f
(
x
)
f(x)
f
(
x
)
is the maximum value taken by any of the given functions
y
y
y
(defined above) at that point
x
x
x
. It is known that
f
(
a
)
=
f
(
b
)
f(a) = f(b)
f
(
a
)
=
f
(
b
)
. Prove that the total length of the intervals in which the function
f
f
f
is increasing is equal to the total length of the intervals in which it is decreasing (that is, both are equal to
(
b
−
a
)
/
2
(b- a)/2
(
b
−
a
)
/2
).(NB Vasiliev)
combinatorics
algebra