Problem 3

Consider

n

disks

C_1,C_2,\ldots,C_n

in the plane such that for each

1\le i<n

, the center of

C_i

lies on the circumference of

C_{i+1}

, and the center of

C_n

lies on the circumference of

C_1

. Define the score to be the number of pairs

(i,j)

for which

C_i

properly contains

C_j

. Determine the maximum possible score.

Step 6 of 6: Give the construction

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\max\operatorname{score}=\frac{(n-1)(n-2)}2

Detailed analysis

For attainment, choose $C_2$ inside $C_1$ , then $C_3$ inside $C_2$ , and so on through $C_{n-1}$ , each with the preceding center on its circumference; choose $C_n$ with center on $C_1$ and circumference through the center of $C_{n-1}$ . Then exactly the pairs $1\le i<j\le n-1$ are containments, giving $\frac{(n-1)(n-2)}2$ .