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MIPT
MIPT Undergraduate Contest 2019
1.5 & 2.5
1.5 & 2.5
Part of
MIPT Undergraduate Contest 2019
Problems
(1)
Scary triginequalinometry on the reals
Source: MIPT Undergraduate Competition 2019 1.5 and 2.5
8/17/2020
Prove the inequality
∑
k
=
1
n
(
x
k
−
x
k
−
1
)
2
≥
4
sin
2
π
2
n
⋅
∑
k
=
0
n
x
k
2
\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2
k
=
1
∑
n
(
x
k
−
x
k
−
1
)
2
≥
4
sin
2
2
n
π
⋅
k
=
0
∑
n
x
k
2
for any sequence of real numbers
x
0
,
x
1
,
.
.
.
,
x
n
x_0, x_1, ..., x_n
x
0
,
x
1
,
...
,
x
n
for which
x
0
=
x
n
=
0.
x_0 = x_n = 0.
x
0
=
x
n
=
0.
inequalities
trigonometry
real analysis
calculus