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I ran into the following problem during some self-motivated studies, and for the last 24 hours I have been unable to solve this problem. The problem arose by itself, meaning it doesn't have a source, like a book or document.

Let $X_t^p = \{ x \in R^n : \sum_{i=1}^n c_i x_i^p = t, \quad \sum_{i=1}^n w_i x_i = W, \quad 0 \leq x_i \leq 1 \quad \forall i = 1,\ldots,n \}$

I need a method that either finds an element $x$ in $X_t^p$, or proves that $X_t^p = \emptyset$.

I'm kind of stuck here. It is part of an optimization problem. I try to maximize $t$ under the condition that $X_t^p \neq \emptyset$, for a fixed $p$.

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