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National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
1965 Dutch Mathematical Olympiad
5
f(x+y)+f(x-y)=2f(x)+2f(y)
f(x+y)+f(x-y)=2f(x)+2f(y)
Source: Netherlands - Dutch NMO 1965 p5
January 31, 2023
algebra
functional
functional equation
Problem Statement
The function ƒ. which is defined for all real numbers satisfies:
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
+
2
f
(
y
)
f(x+y)+f(x-y)=2f(x)+2f(y)
f
(
x
+
y
)
+
f
(
x
−
y
)
=
2
f
(
x
)
+
2
f
(
y
)
Prove that
f
(
0
)
=
0
f(0) = 0
f
(
0
)
=
0
,
f
(
−
x
)
=
f
(
x
)
f(-x) = f(x)
f
(
−
x
)
=
f
(
x
)
,
f
(
2
x
)
=
4
f
(
x
)
f(2x) = 4 f (x)
f
(
2
x
)
=
4
f
(
x
)
,
f
(
x
+
y
+
z
)
=
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
−
f
(
x
)
−
f
(
y
)
−
f
(
z
)
.
f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).
f
(
x
+
y
+
z
)
=
f
(
x
+
y
)
+
f
(
y
+
z
)
+
f
(
z
+
x
)
−
f
(
x
)
−
f
(
y
)
−
f
(
z
)
.
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