Gazeta matematică 2024

Gazeta Matematică 5/2024

P:1791 (Vraja-Lőkös Éva-Ibolya)

Suma a două numere naturale, pare, consecutive este $90$. Aflați produsul acestor numere.

28867 (Natalia Fărcaș)

Fie funcția injectivă 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:\mathbb{R} \to \mathbb{R}} , cu proprietatea că există numerele reale $a$ și $b$ astfel încât $f\left(x\right)\cdot f\left(1-x\right)=f\left(ax+b\right)$ oricare ar fi $x\in \mathbb {R}$.

1. Demonstrați că $f\left(1-b\right)=1$.
2. Dați un exemplu de șir $\left(f_{n}\right)_{n\geq 1}$ de funcții injective $f_{n}:\mathbb {R} \to \mathbb {R}$, 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 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 \log_{n+1} f_n\left(x\right) = a - \log_{n+1} f_n\left(-x\right).}

28868 (Andrei Horvat-Marc)

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

Fie punctele $A\left(2n^{2}+3n,2n+1\right)$, 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\left(2n+1,2n^2+3n\right)} și mulțimea 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} 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.