tmua.me Tau Paper 1 · Question 3

A rectangle has area $1$ . Four circles of positive radii lie inside it, one at each vertex, with each circle tangent to the two sides meeting at that vertex.

Each circle is externally tangent to the two circles at adjacent vertices. The interiors of all four circles are pairwise disjoint.

Write $\lambda=\sqrt{1-1/\sqrt2}$ . What is the greatest possible sum of the areas of the four circles?

4\pi\lambda^2(1-\lambda)^2

\pi\bigl(\lambda^2+(1-\lambda)^2\bigr)^2

4\pi\lambda^4

\dfrac{\pi}{4}

2\pi\bigl(\lambda^4+(1-\lambda)^4\bigr)

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tmua.me Tau Paper 1 · Q3