MathDB
Problems
Contests
National and Regional Contests
Romania Contests
Romania National Olympiad
2005 Romania National Olympiad
1
a little ugly sequences and trig sums
a little ugly sequences and trig sums
Source: Romanian Nationals RMO 2005 - grade 10, problem 1
March 31, 2005
trigonometry
calculus
derivative
algebra proposed
algebra
Problem Statement
Let
n
n
n
be a positive integer,
n
≥
2
n\geq 2
n
≥
2
. For each
t
∈
R
t\in \mathbb{R}
t
∈
R
,
t
≠
k
π
t\neq k\pi
t
=
kπ
,
k
∈
Z
k\in\mathbb{Z}
k
∈
Z
, we consider the numbers
x
n
(
t
)
=
∑
k
=
1
n
k
(
n
−
k
)
cos
(
t
k
)
and
y
n
(
t
)
=
∑
k
=
1
n
k
(
n
−
k
)
sin
(
t
k
)
.
x_n(t) = \sum_{k=1}^n k(n-k)\cos{(tk)} \textrm{ and } y_n(t) = \sum_{k=1}^n k(n-k)\sin{(tk)}.
x
n
(
t
)
=
k
=
1
∑
n
k
(
n
−
k
)
cos
(
t
k
)
and
y
n
(
t
)
=
k
=
1
∑
n
k
(
n
−
k
)
sin
(
t
k
)
.
Prove that if
x
n
(
t
)
=
y
n
(
t
)
=
0
x_n(t) = y_n(t) =0
x
n
(
t
)
=
y
n
(
t
)
=
0
if and only if
tan
n
t
2
=
n
tan
t
2
\tan {\frac {nt}2} = n \tan {\frac t2}
tan
2
n
t
=
n
tan
2
t
. Constantin Buse
Back to Problems
View on AoPS