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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2014 Ukraine Team Selection Test
11
11
Part of
2014 Ukraine Team Selection Test
Problems
(1)
(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x - y) when x + y = u + v
Source: Ukraine TST 2014 p11
4/30/2020
Find all functions
f
:
R
→
R
f: R \to R
f
:
R
→
R
that satisfy the condition
(
f
(
x
)
−
f
(
y
)
)
(
u
−
v
)
=
(
f
(
u
)
−
f
(
v
)
)
(
x
−
y
)
(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)
(
f
(
x
)
−
f
(
y
))
(
u
−
v
)
=
(
f
(
u
)
−
f
(
v
))
(
x
−
y
)
for arbitrary real
x
,
y
,
u
,
v
x, y, u, v
x
,
y
,
u
,
v
such that
x
+
y
=
u
+
v
x + y = u + v
x
+
y
=
u
+
v
.
functional equation
functional
algebra