Sunday Afternoon Maths XXXV

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$$n^2=(10A+B)^2\\=100A+20AB+B^2$$

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$$\frac{1}{2}\times 1\times 1\times \sin(60)\\=\frac{\sqrt{3}}{4}$$ $$\frac{1}{2}ab \sin(60)\\=\frac{ab\sqrt{3}}{4}$$ $$ab=\frac{1}{2}$$ $$L=a^2+b^2-2ab\cos(60)\\ =a^2+\left(\frac{1}{2a}\right)^2-\frac{2a}{2a}\times \frac{1}{2}\\ =a^2+\frac{1}{4a^2}-\frac{1}{2}$$ $$0=\frac{dL}{da}=2a-\frac{1}{2a^3}\\ 4a^4-1=0\\ a^4=\frac{1}{4}\\ a=\frac{1}{\sqrt{2}}$$

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$$\mathbb{P}(M_2=\mathrm{blue})=\mathbb{P}(M_2=\mathrm{blue}|M_1=\mathrm{blue})\mathbb{P}(M_1=\mathrm{blue})\\+\mathbb{P}(M_2=\mathrm{blue}|M_1=\mathrm{yellow})\mathbb{P}(M_1=\mathrm{yellow})\\ =\left(\frac{m+k}{m+n+k}\right)\left(\frac{m}{m+n}\right)+\left(\frac{m}{m+n+k}\right)\left(\frac{n}{m+n}\right)\\ =\frac{m(m+k)+mn}{(m+n)(m+n+k)}\\ =\frac{m(m+n+k)}{(m+n)(m+n+k)}\\ =\frac{m}{m+n} $$

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$$\frac{111n}{3n}=\frac{111}{3}\\=37$$

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