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National and Regional Contests
China Contests
China Team Selection Test
2019 China Team Selection Test
5
Weird FE on Rationals
Weird FE on Rationals
Source: China TST Test 1 Day 2 Q5
March 11, 2019
functional equation
function
China TST
Problem Statement
Determine all functions
f
:
Q
→
Q
f: \mathbb{Q} \to \mathbb{Q}
f
:
Q
→
Q
such that
f
(
2
x
y
+
1
2
)
+
f
(
x
−
y
)
=
4
f
(
x
)
f
(
y
)
+
1
2
f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}
f
(
2
x
y
+
2
1
)
+
f
(
x
−
y
)
=
4
f
(
x
)
f
(
y
)
+
2
1
for all
x
,
y
∈
Q
x,y \in \mathbb{Q}
x
,
y
∈
Q
.
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