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Given that cos(11*pi/24) = 1 / sqrt(1 + (a + b*sqrt(3) + sqrt(c + d*sqrt(3)))²), where a, b, c, d are natural numbers, find the least possible value of a + b + c + d.

1. 14
2. 15
3. 17
4. 19

Correct answer: 15

Solution

tan(11*pi/24) = tan(82.5 deg) = cot(7.5 deg) = 2 + sqrt(3) + sqrt(8 + 4*sqrt(3)), giving a=2, b=1, c=8, d=4, so a+b+c+d = 15.

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