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For positive integer \(n\), define $$f(n) = n + \frac{16+5n-3n^2}{4n+3n^2} + \frac{32+n-3n^2}{8n+3n^2}+\frac{48-3n-3n^2}{12n+3n^2}+\cdots + \frac{25n-7n^2}{7n^2}.$$ Then, evaluate \(\lim\limits_{n\to \infty}f(n)\).
Published:
Let \(\overline{BD}\) and \(\overline{CE}\) be the altitudes of acute triangle \(ABC\). Let \(\omega_B\) and \(\omega_C\) be the circles with diameter \(\overline{BD}\) and \(\overline{CE}\), respectively. Suppose \(\overline{BD}\) intersect \(\omega_C\) at \(P\), \(\overline{CE}\) intersect \(\omega_B\) at \(R\), \(\omega_B\) and \(\omega_C\) intersects at \(X\) and \(Y\). If \(\overline{XY}\) intersect \(\overline{DE}\) at \(T\), prove that \(\overline{TP}=\overline{TR}\).