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National and Regional Contests
Belgium Contests
Flanders Math Olympiad
2021 Flanders Math Olympiad
4
4
Part of
2021 Flanders Math Olympiad
Problems
(1)
f(x/(x^2 + x + 1))=x^2/(x^4 + x^2 + 1)
Source: Flanders Math Olympiad 2021 p4
12/24/2022
(a) Prove that for every
x
∈
R
x \in R
x
∈
R
holds that
−
1
≤
x
x
2
+
x
+
1
≤
1
3
-1 \le \frac{x}{x^2 + x + 1} \le \frac 13
−
1
≤
x
2
+
x
+
1
x
≤
3
1
(b) Determine all functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
for which for every
x
∈
R
x \in R
x
∈
R
holds that
f
(
x
x
2
+
x
+
1
)
=
x
2
x
4
+
x
2
+
1
f \left( \frac{x}{x^2 + x + 1} \right) = \frac{x^2}{x^4 + x^2 + 1}
f
(
x
2
+
x
+
1
x
)
=
x
4
+
x
2
+
1
x
2
inequalities
functional
algebra
functional equation