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Equation with powers of 3 and trig

Source: Bulgarian Spring Tournament 2023 11.1

March 25, 2023
trigonometryalgebra

Problem Statement

Find all real aa such that the equation 3cos⁡(2x)+1−(a−5)3cos⁡2(2x)=73^{\cos (2x)+1}-(a-5)3^{\cos^2(2x)}=7 has a real root.
This was the statement given at the contest, but there was actually a typo and the intended equation was 3cos⁡(2x)+1−(a−5)3cos⁡2(x)=73^{\cos (2x)+1}-(a-5)3^{\cos^2(x)}=7, which is much easier.
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