Problem Analysis #26

[THOUGHT] The circles $O_n$ (including $O_0$) are all tangent to the two intersecting lines $y=-1$ and $PQ$. In an angle of half-angle $\alpha$, consecutive circles tangent to both sides and to each other have radii ratio

Given $\sum_{n\ge1}\pi r_n^2=\frac{\pi}{8}$ and $r_0=1$, we have $r_n=t^n$, so

Thus $\frac{1-\sin\alpha}{1+\sin\alpha}=\frac13\Rightarrow \sin\alpha=\frac12\Rightarrow \alpha=\frac{\pi}{6}$, so the angle between the lines is $2\alpha=\frac{\pi}{3}$ (60°). Therefore the tangent line $PQ$ makes a 60° angle with the horizontal $y=-1$, so its slope has magnitude $\tan60=\sqrt3$. The tangent line at $P(x_0,y_0)$ has slope $-x_0/y_0$, hence $|x_0/y_0|=\sqrt3$. The configuration inside triangle $P'PQ$ forces the interior angle at $Q$ to be 60°, giving $y_0>0$, so $x_0/y_0=\sqrt3\Rightarrow y_0=x_0/\sqrt3$. With $x_0^2+y_0^2=1$, we get $x_0^2+x_0^2/3=1\Rightarrow x_0^2=3/4$ and $y_0=1/2$. Compute $2024\cdot x_0^2\cdot y_0$. [/THOUGHT] [PYTHON] from fractions import Fraction x0_sq = Fraction(3,4) y0 = Fraction(1,2) value = 2024 * x0_sq * y0 print(value) print(int(value)) [/PYTHON]

[THOUGHT] $\boxed{759}$