Question:

1. Supposed that the Greek mathematician Diophantus passed 1/6 of his life in childhood, 1/12 in youth and 1/7 as an adult bachelor. Five years after his marriage, a son was born. The son died four years before his father’s death at half his father’s final age. What was Diophantus’ age at his death?
2. Find all solutions of the equation \(\sin x+3\sin 2x=0\) on the interval \(0\le x\le 2\pi \)
3. The polynomial \({{a}^{3}}{{x}^{2}}-14ax-24\) is divisible by \(\left( x-2 \right)\) and \(\left( x-t \right)\). Find all the possible values of t .
4. If \(f\) is a function such that \(4f\left( {{x}^{-1}}+1 \right)+8f\left( x+1 \right)={{\log }_{12}}x\) , then what is
   \[f\left( 10 \right)+f\left( 13 \right)+f\left( 17 \right)\]
5. If \(x+\frac{1}{x}=2\), and \(f\left( t \right)={{x}^{t}}+\frac{1}{{{x}^{t}}}\), then what is \(\sum\limits_{k=0}^{2000}{f\left( {{2}^{k}} \right)}\) ?

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