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Hungary Contests
Kürschák Math Competition
1979 Kurschak Competition
2
2
Part of
1979 Kurschak Competition
Problems
(1)
f(x) = x if f(x) <= x and f(x + y) <= f(x) + f(y)
Source: 1979 Hungary - Kürschák Competition p2
10/15/2022
f
f
f
is a real-valued function defined on the reals such that
f
(
x
)
≤
x
f(x) \le x
f
(
x
)
≤
x
and
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
f(x + y) \le f(x) + f(y)
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
for all
x
,
y
x, y
x
,
y
. Prove that
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
for all
x
x
x
.
algebra
functional
Functional inequality
function