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Ceiling of power of root of cubic equation is divisible by 3

Source: Germany 2023, Problem 6

June 16, 2023
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Problem Statement

The equation x3−3x2+1=0x^3-3x^2+1=0 has three real solutions x1<x2<x3x_1<x_2<x_3. Show that for any positive integer nn, the number ⌈x3n⌉\left\lceil x_3^n\right\rceil is a multiple of 33.
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