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MATH / For 1≤i≤215 let a_(i)=(1)/(2^(i)) and a₂₁₆=(1)/(2²¹⁵). Let x₁,x₂,…,x₂₁₆ be positive real numbers such that ∑_(i=1)²¹⁶x_(i)=1 and …

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problem

For 1≤i≤2151 \leq i \leq 215 let ai=12ia_i = \dfrac{1}{2^{i}} and a216=12215a_{216} = \dfrac{1}{2^{215}}. Let x1,x2,…,x216x_1, x_2, \dots, x_{216} be positive real numbers such that ∑i=1216xi=1\sum_{i=1}^{216} x_i=1 and ∑1≤i<j≤216xixj=107215+∑i=1216aixi22(1−ai).\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}.Find the maximum possible value of x2.x_2.
Plain-text mathematical notation (without MathML)
For 1≤i≤215 let a_(i)=(1)/(2^(i)) and a₂₁₆=(1)/(2²¹⁵). Let x₁,x₂,…,x₂₁₆ be positive real numbers such that ∑_(i=1)²¹⁶x_(i)=1 and
∑_(1≤i<j≤216)x_(i)x_(j)=(107)/(215)+∑_(i=1)²¹⁶(a_(i)x_(i)²)/(2(1−a_(i))).Find the maximum possible value of x₂.
Original LaTeX notation
For $1 \leq i \leq 215$ let $a_i = \dfrac{1}{2^{i}}$ and $a_{216} = \dfrac{1}{2^{215}}$. Let $x_1, x_2, \dots, x_{216}$ be positive real numbers such that $\sum_{i=1}^{216} x_i=1$ and
\[\sum_{1 \leq i < j \leq 216} x_ix_j = \dfrac{107}{215} + \sum_{i=1}^{216} \dfrac{a_i x_i^{2}}{2(1-a_i)}.\]Find the maximum possible value of $x_2.$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
27469141-959f-590b-ac7d-56a6fa50885f
task_key
intermediate~5falgebra--test--27469141-959f-590b-ac7d-56a6fa50885f
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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