MathDB

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Part of 2021 Middle European Mathematical Olympiad

Problems(2)

Simple sequences

Source: 2021 MEMO I-1

9/5/2021
Determine all real numbers A such that every sequence of non-zero real numbers x1,x2,…x_1, x_2, \ldots satisfying xn+1=A−1xn x_{n+1}=A-\frac{1}{x_n} for every integer n≥1n \ge 1, has only finitely many negative terms.
SequencealgebramemoMEMO 2021constant
Functional Inequality (is back)

Source: 2021 MEMO T-1

9/5/2021
Determine all functions f:R→Rf: \mathbb{R} \to \mathbb{R} such that the inequality f(x2)−f(y2)≤(f(x)+y)(x−f(y)) f(x^2)-f(y^2) \le (f(x)+y)(x-f(y)) holds for all real numbers xx and yy.
functional equationFunctional inequalityalgebramemoMEMO 2021