Problem:
There is a unique angle θ between 0∘ and 90∘ such that for nonnegative integers n, the value of tan(2nθ) is positive when n is a multiple of 3, and negative otherwise. The degree measure of θ is qp, where p and q are relatively prime integers. Find p+q.
Solution:
Suppose tanθ>0. Then θ must lie in the first or third quadrant, so θ∈(0∘,90∘)(mod180∘).
For tan(2nθ) to be positive for all n divisible by 3, each angle 2nθ (where n≡0(mod3)) must lie in the same quadrant as θ modulo 180∘.
This implies 8θ≡θ(mod180∘), so 7θ≡0(mod180∘). Hence, θ=7180∘⋅k for some integer k.
To find values of θ in (0∘,90∘), we try k=1,2,3:
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For k=1, θ=7180∘≈25.7∘. Then 2θ≈51.4∘, which is in the first quadrant, so tan(2θ)>0.
But 4θ≈102.9∘, which is in the second quadrant, where tangent is negative. Discard.
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For k=2, θ=7360∘≈51.4∘. Then 4θ≈205.7∘, which is in the third quadrant, so tan(4θ)>0, but 8θ≈411.4∘ lies in the second quadrant, so tan(8θ)<0. Discard.
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For k=3, θ=7540∘≈77.1∘. Then 2θ≈154.3∘, 4θ≈308.6∘, 8θ≈617.1∘ — all of which lie in the third quadrant modulo 180∘, so their tangents are positive.
Thus, θ=7540∘ is the only valid solution. The answer is 540+7=547.
The problems on this page are the property of the MAA's American Mathematics Competitions