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Maximum number of real solutions Aslı can guarantee

Source: 2023 Turkey TST D3 P8

March 29, 2023
algebra

Problem Statement

Initially the equation ⋆1x−1⋆1x−2⋆1x−4...⋆1x−22023=0\star \frac{1}{x-1} \star \frac{1}{x-2} \star \frac{1}{x-4} ... \star \frac{1}{x-2^{2023}}=0
is written on the board. In each turn Aslı and Zehra deletes one of the stars in the equation and writes ++ or −- instead. The first move is performed by Aslı and continues in order. What is the maximum number of real solutions Aslı can guarantee after all the stars have been replaced by signs?
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