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The paper generalizes a problem of the so-called non-discrete silent duels, which were considered by Lang and Kimeldorf in [6]. It contains a solution of the following class of zero-sum two-person games. Let- for i = 1 and 2, a finite nonnegative number M
i- and a nondecreasing, continuous function Ai(t) from [0, 1] onto [0, 1] be given. A pure strategy for player i is any measure μi on [0, 1] satisfying μi[0,1] = M
i. The payoff function it is defined as .
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