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Problems
Contests
National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2024 Turkey Team Selection Test
6
6
Part of
2024 Turkey Team Selection Test
Problems
(1)
n-variable inequality from Turkey TST
Source: 2024 Turkey TST P6
3/18/2024
For a positive integer
n
n
n
and real numbers
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \dots ,a_n
a
1
,
a
2
,
…
,
a
n
we'll define
b
1
,
b
2
,
…
,
b
n
+
1
b_1, b_2, \dots ,b_{n+1}
b
1
,
b
2
,
…
,
b
n
+
1
such that
b
k
=
a
k
+
max
(
a
k
+
1
,
a
k
+
2
)
b_k=a_k+\max({a_{k+1},a_{k+2}})
b
k
=
a
k
+
max
(
a
k
+
1
,
a
k
+
2
)
for all
1
≤
k
≤
n
1\leq k \leq n
1
≤
k
≤
n
and
b
n
+
1
=
b
1
b_{n+1}=b_1
b
n
+
1
=
b
1
. (Also
a
n
+
1
=
a
1
a_{n+1}=a_1
a
n
+
1
=
a
1
and
a
n
+
2
=
a
2
a_{n+2}=a_2
a
n
+
2
=
a
2
) Find the least possible value of
λ
\lambda
λ
such that for all
n
,
a
1
,
…
,
a
n
n, a_1, \dots, a_n
n
,
a
1
,
…
,
a
n
the inequality
λ
[
∑
i
=
1
n
(
a
i
−
a
i
+
1
)
2024
]
≥
∑
i
=
1
n
(
b
i
−
b
i
+
1
)
2024
\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}
λ
[
i
=
1
∑
n
(
a
i
−
a
i
+
1
)
2024
]
≥
i
=
1
∑
n
(
b
i
−
b
i
+
1
)
2024
holds.
inequalities