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Found problems: 2

2017 Korea National Olympiad, problem 4
Tags: sequence , algebraic number , function , algebra
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as \[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \] Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.

2005 Miklós Schweitzer, 7
Tags: algebra , symmetric , algebraic number
Let $t\in R$. Prove that $\exists A:R \times R \to R$ such that A is a symmetric, biadditive, nonzero function and $A(tx,x)=0 \,\forall x\in R$ iff t is transcendental or (t is algebraic and t,-t are conjugates over $\mathbb{Q}$).