MathDB

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Part of 2017 VJIMC

Problems(2)

Power Series with special coefficients is rational function

Source: VJIMC 2017, Category II, Problem 1

4/2/2017
Let (an)n=1∞(a_n)_{n=1}^{\infty} be a sequence with an∈{0,1}a_n \in \{0,1\} for every nn. Let F:(−1,1)→RF:(-1,1) \to \mathbb{R} be defined by F(x)=∑n=1∞anxnF(x)=\sum_{n=1}^{\infty} a_nx^n and assume that F(12)F\left(\frac{1}{2}\right) is rational. Show that FF is the quotient of two polynomials with integer coefficients.
algebrapolynomialrational functionfunction
Functional equation with continuity implies constant

Source: VJIMC 2017, Category I, Problem 1

4/2/2017
Let f:R→Rf:\mathbb{R} \to \mathbb{R} be a continuous function satisfying f(x+2y)=2f(x)f(y)f(x+2y)=2f(x)f(y) for every x,y∈Rx,y \in \mathbb{R}. Prove that ff is constant.