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n-variable inequality from Turkey TST

Source: 2024 Turkey TST P6

March 18, 2024

inequalities

Problem Statement

For a positive integer

n

and real numbers

a_1, a_2, \dots ,a_n

we'll define

b_1, b_2, \dots ,b_{n+1}

such that

b_k=a_k+\max({a_{k+1},a_{k+2}})

for all

1\leq k \leq n

and

b_{n+1}=b_1

. (Also

a_{n+1}=a_1

and

a_{n+2}=a_2

) Find the least possible value of

\lambda

such that for all

n, a_1, \dots, a_n

the inequality

\lambda \Biggl[ \sum_{i=1}^n(a_i-a_{i+1})^{2024} \Biggr] \geq \sum_{i=1}^n(b_i-b_{i+1})^{2024}

holds.

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