Gazeta matematică 2012

Gazeta Matematică 3/2012

E:14309 (Alexandru Vele)

Determinați numerele naturale 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_1, a_2, a_3, a_4, a_5, a_6, a_7} astfel încât să avem egalitatea:

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 2012 = a_1 \cdot3^x + a_2\cdot3^y + a_3\cdot3^z + a_4\cdot3^t + a_5\cdot3^u + a_6\cdot3^r + a_7\cdot3^s.}

Arătați 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 a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 = m^2 + n^2, m,n\in\Nu}

Alexandru Vele, Târgu Lăpuș

Soluție. Dacă 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_1, a_2, a_3, a_4, a_5, a_6, a_7} sunt mai mici decât 3, atunci $a_{1}\cdot 3^{x}+a_{2}\cdot 3^{y}+a_{3}\cdot 3^{z}+a_{4}\cdot 3^{t}+a_{5}\cdot 3^{u}+a_{6}\cdot 3^{r}+a_{7}\cdot 3^{s}$ poate fi privită ca scrierea în baza 3 a lui 2012. Cum $2012=2\cdot 3^{0}+1\cdot 3^{1}+1\cdot 3^{2}+2\cdot 3^{3}+0\cdot 3^{4}+2\cdot 3^{5}.+2\cdot 3^{6}$ avem $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}=2+1+1+2+0+2+2=10=1^{2}+3^{2}.$ Dacă cel puțin unul dintre numerele $a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7}$ este mai mare sau egal cu 3, atunci problema nu mai rămâne adevărată; 2012 se poate scrie ca o suma de puteri ale lui 3, dar suma a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 nu se mai scrie, sigur, ca sumă de două pătrate.

Gazeta Matematică 3/2012

E:14310 (Traian Covaciu)

Fie $n,n+2,n+6$ trei numere naturale și $S$ suma lor.

a) Dați exemplu de cel puțin trei valori pentru $n\in \mathbb {N}$ astfel încât numerele $n,n+2,n+6$ să fie simultan numere prime.

b) Dacă $n,n+2,n+6$ sunt simultan numere prime, arătați că există 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 k \in \mathbb{N} } 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 S = 9k + 5} .

c) Dacă 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, n + 2, n + 6 } sunt numere prime, determinați restul împărțirii numărului 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 S} la 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 18 } .

E:14312 (Iulian Bunu) Andrei are o anumită sumă de bani și se pregătește pentru două evenimente: Tabăra de Matematică și aniversarea Anei. Dacă ar câștiga premiul de 50 de lei și n-ar putea merge la aniversare, noua sumă ar fi cubul unui număr natural, iar dacă n-ar câștiga nimic, dar ar cheltui pentru cadou 50 de lei, noua sumă ar fi pătratul aceluiași număr natural. Ce sumă are Andrei?

E:14313 (Emil Florin Bizău și Ioan Bizău)

Determinați numerele întregi 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} , 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 y} 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 z} 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 \frac{2x + 3}{3} = \frac{2}{3y - 1} = \frac{5}{4z - 3}} .

Gazeta Matematică 4/2012

E:14331 (Cristina Vijdeluc și Mihai Vijdeluc)

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 \ge2 } un număr natural. Arătați că numărul 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^4 + n^2 + 3 } nu poate fi scris ca sumă a două numere prime.

E:14336 (Gheorghe Szöllösy)

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 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} două numere reale nenule, fixate. Determinați toate 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 : \mathbb{R} \to \mathbb{R}} cu proprietatea: 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(x) - f(y) = (ax + by)f(x)f(y),}
pentru orice 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} 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 y} numere reale.

Gazeta Matematică 9/2012

E:14380 (Vasile Ienuțaș)

Determinați cifrele 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} știind 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 \overline{ab}=(a+b)(a+b-1)} .

E:14383 (Gheorghe Gherasim)

Numerele naturale distincte 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} , 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} verifică 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 9 \cdot [\,a, b]\,=a \cdot b \cdot (\,a \cdot b)\,} .

i) Arătați 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 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} nu sunt prime între ele.

ii) Arătați că diferența numerelor este cel puțin 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 3} .

Se consideră 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 [a,b]} reprezintă cel mai mic multiplu comun al numerelor 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} , iar 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,b)} este cel mai mare divizor comun al numerelor 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} .