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2013 Turkey Team Selection Test
2
f(x^2) = f(x)^2 - 2xf(x), f(x)=f(x-1), 1<x<y ==> f(x)<f(y)
f(x^2) = f(x)^2 - 2xf(x), f(x)=f(x-1), 1<x<y ==> f(x)<f(y)
Source: Turkey TST 2013 - Day 3 - P2
April 2, 2013
function
induction
algebra proposed
algebra
Problem Statement
Determine all functions
f
:
R
→
R
+
f:\mathbf{R} \rightarrow \mathbf{R}^+
f
:
R
→
R
+
such that for all real numbers
x
,
y
x,y
x
,
y
the following conditions hold:
i
.
f
(
x
2
)
=
f
(
x
)
2
−
2
x
f
(
x
)
i
i
.
f
(
−
x
)
=
f
(
x
−
1
)
i
i
i
.
1
<
x
<
y
⟹
f
(
x
)
<
f
(
y
)
.
\begin{array}{rl} i. & f(x^2) = f(x)^2 -2xf(x) \\ ii. & f(-x) = f(x-1)\\ iii. & 1<x<y \Longrightarrow f(x) < f(y). \end{array}
i
.
ii
.
iii
.
f
(
x
2
)
=
f
(
x
)
2
−
2
x
f
(
x
)
f
(
−
x
)
=
f
(
x
−
1
)
1
<
x
<
y
⟹
f
(
x
)
<
f
(
y
)
.
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