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Home » Circumference = \(\pi \times\) diameter. - Decision Point

Circumference = \(\pi \times\) diameter. - Decision Point

📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 We want the probability of exactly 2 successes in 3 trials:

📌 But because the arrangement is circular, we must be careful with wrap-around cases (i.e., positions 5,6,1,2 or 6,1,2,3 possible).

📌 Because this is complex for a math olympiad and must have a clean answer, we consider symmetry and known results or constructive counting.

📌 But given time and olympiad style, and since the circle wraps, we proceed with case analysis on placements of AAAA.

📌 Possible placements of 4 consecutive As in a circle of 6:

📌 But overlaps make inclusion-exclusion messy.

📌 We use inclusion-exclusion with care.

📌 But for olympiad, a clean trick: use the fact that the total number of binary circular strings of length 6 is 64, and due to symmetry and small size, the number

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= 4 egin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix}
+ 4 egin{vmatrix} -6 & 9 \ 4 & -18 \end{vmatrix}
Compute minors:
$ egin{vmatrix} 9 & -18 \ -18 & 20 \end{vmatrix} = (9)(20) - (-18)^2...
$ egin{vmatrix} -6 & -18 \ 4 & 20 \end{vmatrix} = (-6)(20) - (-18)(4)...
$ egin{vmatrix} -6 & 9 \ 4 & -18 \end{vmatrix} = (-6)(-18) - (9)(4) =...
Now plug into determinant expression:
4(-144) + 6(-48) + 4(72) = -576 - 288 + 288 = -576
Since the determinant is non-zero ($ -576
eq 0 $), the conic is **non-degenerate**, despite the discriminant bei...

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