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National and Regional Contests
Latvia Contests
Latvia BW TST
2018 Latvia Baltic Way TST
P4
P4
Part of
2018 Latvia Baltic Way TST
Problems
(1)
Functional inequality with size constraint
Source: 2018 Latvia BW TST P4
6/4/2022
Let
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
be a function that satisfies
2
f
(
x
)
−
2
f
(
x
)
−
f
(
2
x
)
≥
2
\sqrt{2f(x)}-\sqrt{2f(x)-f(2x)}\ge 2
2
f
(
x
)
−
2
f
(
x
)
−
f
(
2
x
)
≥
2
for all real
x
x
x
. Prove for all real
x
x
x
: (a)
f
(
x
)
≥
4
f(x)\ge 4
f
(
x
)
≥
4
; (b)
f
(
x
)
≥
7.
f(x)\ge 7.
f
(
x
)
≥
7.
function
inequalities
algebra
algebra unsolved