Question 2
In the plane a circle $(C)$ of radius $R$ contains and is internally tangent to a circle $(C')$ of radius $R/2$. Let $H$ be the family of circles inside $(C)$, outside $(C')$, and tangent to both. For each integer $n\ge3$ and positive real numbers $p_1,p_n$, prove that
$$(p_1-p_n)^2=(n-1)^2\bigl(2(p_1+p_n)-(n-1)^2-8\bigr)$$
is necessary and sufficient for there to be $n$ distinct circles $(C_1),\ldots,(C_n)$ in $H$ such that $(C_i)$ is externally tangent to $(C_{i-1})$ and $(C_{i+1})$ for $i=2,\ldots,n-1$, with radii $R/p_1$ and $R/p_n$ for $(C_1)$ and $(C_n)$ respectively.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.
$$(p_1-p_n)^2=(n-1)^2\bigl(2(p_1+p_n)-(n-1)^2-8\bigr)$$
is necessary and sufficient for there to be $n$ distinct circles $(C_1),\ldots,(C_n)$ in $H$ such that $(C_i)$ is externally tangent to $(C_{i-1})$ and $(C_{i+1})$ for $i=2,\ldots,n-1$, with radii $R/p_1$ and $R/p_n$ for $(C_1)$ and $(C_n)$ respectively.
The supplied source has no question point values. Each problem uses 1 editorial point for import compatibility; this is not an official scoring scale.