Gazeta matematică 2024

Gazeta Matematică 5/2024

P:1791 (Vraja-Lőkös Éva-Ibolya)

Suma a două numere naturale, pare, consecutive este Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 90} . Aflați produsul acestor numere.

P:1792 (Monica Dragoș)

Determinați numărul natural Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{ab}} pentru care Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \times \overline{b00b} + \overline{aa} = 2024} .

28867 (Natalia Fărcaș)

Fie funcția injectivă $f:\mathbb {R} \to \mathbb {R}$, cu proprietatea că există numerele reale Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} astfel încât Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(x\right) \cdot f\left(1-x\right) = f\left(ax+b\right)} oricare ar fi Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in \mathbb{R}} .

1. Demonstrați că Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left(1-b\right)=1} .
2. Dați un exemplu de șir $\left(f_{n}\right)_{n\geq 1}$ de funcții injective Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_n:\mathbb{R} \to \mathbb{R}} , cu proprietatea că există $a,b\in \mathbb {R}$, astfel încât pentru orice $x\in \mathbb {R}$, avem
   $$f_{n}\left(x\right)\cdot f_{n}\left(1-x\right)=f_{n}\left(ax+b\right)$$
   și
   $$\log _{n+1}f_{n}\left(x\right)=a-\log _{n+1}f_{n}\left(-x\right).$$

28868 (Andrei Horvat-Marc)

Fie $n\in \mathbb {N^{\ast }}$ și funcțiile $f:\left[0,2n^{2}+3n\right]\to \left[1,2n+1\right]$, $f\left(x\right)={\frac {{\sqrt {8x+9}}-1}{2}}$ și $g:\left[1,2n+1\right]\to \left[0,2n^{2}+3n\right]$, $g\left(x\right)=f^{-1}\left(x\right)$.

Fie punctele $A\left(2n^{2}+3n,2n+1\right)$, $B\left(2n+1,2n^{2}+3n\right)$ și mulțimea $M$ a punctelor din plan cuprinse între graficele funcțiilor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} și Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} și dreapta Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle AB} . Aflați numărul punctelor din Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} care au ambele coordonate întregi.