Tübitak Ortaokul 1. Aşama 2024 Soru 19

[matematikolimpiyati]

$f(x)=\dfrac{x^3}{3x^2-3x+1}$ olmak üzere, $$f \left( \dfrac{1}{33} \right) + f \left( \dfrac{2}{33} \right) + f \left( \dfrac{3}{33} \right) + \cdots + f \left( \dfrac{31}{33} \right) + f \left( \dfrac{32}{33} \right)$$ toplamı kaçtır?

$\textbf{a)}\ \dfrac{65}{33}  \qquad\textbf{b)}\ \dfrac{31}{2}  \qquad\textbf{c)}\ 16  \qquad\textbf{d)}\ \dfrac{496}{33}  \qquad\textbf{e)}\ \dfrac{33}{2}$

[ibka00]

Cevap ektedir.

[geo]

Yanıt: $\boxed C$

$f(x) = \dfrac {x^3}{3x^2 - 3x + 1} = \dfrac {x^3}{(1-x)^3 + x^3}$ ve $f(1-x) = \dfrac {(1-x)^3}{x^3 + (1-x)^3}$ olduğu için $f(x) + f(1-x)=1$.

$S = f \left( \dfrac{1}{33} \right) + f \left( \dfrac{2}{33} \right) + f \left( \dfrac{3}{33} \right) + \cdots + f \left( \dfrac{31}{33} \right) + f \left( \dfrac{32}{33} \right) $

$S = f \left( \dfrac{32}{33} \right) + f \left( \dfrac{31}{33} \right) + f \left( \dfrac{30}{33} \right) + \cdots + f \left( \dfrac{2}{33} \right) + f \left( \dfrac{1}{33} \right) $

Taraf tarafa toplarsak, $2S = 32 \cdot 1 = 32 \Longrightarrow S = 16$.