MathDB
Problems
Contests
International Contests
IMO Longlists
1979 IMO Longlists
65
Functional Equation - prove equality from inequality
Functional Equation - prove equality from inequality
Source: ILL 1979 - Problem 65.
June 5, 2011
inequalities
function
algebra proposed
algebra
Problem Statement
Given a function
f
f
f
such that
f
(
x
)
≤
x
∀
x
∈
R
f(x)\le x\forall x\in\mathbb{R}
f
(
x
)
≤
x
∀
x
∈
R
and
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
{
x
,
y
}
∈
R
f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}
f
(
x
+
y
)
≤
f
(
x
)
+
f
(
y
)
∀
{
x
,
y
}
∈
R
, prove that
f
(
x
)
=
x
∀
x
∈
R
f(x)=x\forall x\in\mathbb{R}
f
(
x
)
=
x
∀
x
∈
R
.
Back to Problems
View on AoPS