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Polynomial with f(x) f(1/x) = f(x) + f(1/x)

This type of question, asked 2 times: 28 Jan 2025, 8 Apr 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    28 Jan 2025

    145
    Let f:R−{0}→(−∞,1)f\mathbf{:R - \{ 0\} \rightarrow ( - \infty,1)} be a polynomial of degree 2 , satisfying f(x)f(1x)=f(x)+f(1x)f(x)f\left( \frac{1}{x} \right)=f(x) +f\left( \frac{1}{x} \right). If f( K)=−2 Kf(\text{ }K) = - 2\text{ }K, then the sum of squares of all possible values of K is :

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  2. Question 2

    8 Apr 2026 S2

    75
    Let f be a polynomial function such that log⁡2(f(x))=(log⁡2(2+23+29+...∞))\log_{2}(f(x)) = \left( \log_{2}\left( 2 + \frac{2}{3} + \frac{2}{9} + ...\infty \right) \right). log⁡3(1+f(x)f(1/x)),x>0\log_{3}\left( 1 + \frac{f(x)}{f(1/x)} \right),x > 0 and f(6)=37f(6) = 37. Then ∑n=110f(n)\sum_{n = 1}^{10} f(n) is equal to ____\_\_\_\_ .