28868: Difference between revisions
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Între segmentele <math>G_f</math> și <math>G_g</math> se situează și punctul <math>Q\left(1,1\right)</math>, însă considerăm <math>M</math> ca fiind mulțimea închisă delimitată de <math>G_f</math>, <math>G_g</math> și <math>\left[AB\right]</math>. | Între segmentele <math>G_f</math> și <math>G_g</math> se situează și punctul <math>Q\left(1,1\right)</math>, însă considerăm <math>M</math> ca fiind mulțimea închisă delimitată de <math>G_f</math>, <math>G_g</math> și <math>\left[AB\right]</math>. | ||
Fie punctele <math>C\left(2n^2+3n,2n^2+3n\right)</math>, <math>E\left(2,2n^2+3n\right)</math>, <math>F\left(2n^2+3n,2\right)</math>. Observăm că, datorită simetriei, triunghiurile curbilinii <math>DBE</math> și <math>DAF</math> conțin același număr de puncte laticeale. Notăm <math> | Fie punctele <math>C\left(2n^2+3n,2n^2+3n\right)</math>, <math>E\left(2,2n^2+3n\right)</math>, <math>F\left(2n^2+3n,2\right)</math>. Observăm că, datorită simetriei, triunghiurile curbilinii <math>DBE</math> și <math>DAF</math> conțin același număr de puncte laticeale. | ||
Notăm cu <math>A_n</math> numărul punctelor laticeale din interiorul și de pe frontiera pătratului <math>DFCE</math>., cu <math>T_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului <math>CAB</math> și cu <math> | |||
S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>. | |||
Avem <math>A_n = \left(2n^2+3n-1\right)^2</math> | Avem <math display="block">A_n = \left(2n^2+3n-1\right)^2,</math> <math display="block">T_n = \sum\limits_{k=1}^{2n^2+n} k = \dfrac{1}{2}n\left(2n+1\right)\left(2n^2+n+1\right), </math> și <math display="block">S_n = \sum\limits_{k=2}^{2n+1} \left(2n^2+3n+1-g\left(k\right)\right) = \dfrac{1}{3}n\left(2n+1\right)\left(4n+1\right).</math>Atunci <math>M_n = A_n - 2S_n -T_n+3</math>, în formula precedenă de adaugă <math>3</math> pentru a corecta faptul că punctele <math>A</math>, <math>B</math>, respectiv <math>D</math> sunt puncte comune ale regiunilor <math>ADF</math>, <math>BDE</math>, respectiv <math>CAB</math>. Se obține | ||
Atunci <math>M_n = A_n - 2S_n -T_n+3</math>, în formula precedenă de adaugă <math>3 | |||
<math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math> | <math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math> | ||
Cazuri particulare: <math>M_1 = 3</math> este ușor de construit și verificat, <math>M_2 = 57</math> este reprezentat în figura de mai sus, <math>M_3 = 266 </math> și <math>M_4 = 778</math>. | Cazuri particulare: <math>M_1 = 3</math> este ușor de construit și verificat, <math>M_2 = 57</math> este reprezentat în figura de mai sus, <math>M_3 = 266 </math> și <math>M_4 = 778</math>. | ||
Revision as of 08:07, 4 August 2025
28868 (Andre 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 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\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.
Soluție.
Cum 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 = f^{-1}} , se obține că funcția 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]} este definită prin 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) = \dfrac{\left(x-1\right)\left(x+2\right)}{2}} . Mai mult, 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(k\right) \in \mathbb{N}} 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 k \in \left[1, 2n+1\right] \cap \mathbb {N}} .
Avem 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) \in G_g} , 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\left(2n^2+3n,2n+1\right) \in G_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_f \cap G_g = \left\{D\left(2,2\right)\right\}} .
Au loc inegalitățile 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) \le x} 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 \left[2,2n^2+3n\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 g\left(x\right) \ge x} 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 \left[2, 2n+1\right]} .
Considerăm că 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} este mulțimea tuturor punctelor din plan cuprinse în interiorul triunghiului curbiliniu 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 ABD} , deci este necesar să numărăm punctele laticeale din interiorul triunghiului curbiliniu 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 ABD} , vom nota cu 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_n} acest număr.
Între segmentele 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_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_g} se situează și punctul 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 Q\left(1,1\right)} , însă considerăm 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} ca fiind mulțimea închisă delimitată de 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_f} , 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_g} 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 \left[AB\right]} .
Fie punctele 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 C\left(2n^2+3n,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 E\left(2,2n^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 F\left(2n^2+3n,2\right)} . Observăm că, datorită simetriei, triunghiurile curbilinii 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 DBE} 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 DAF} conțin același număr de puncte laticeale.
Notăm cu 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_n} numărul punctelor laticeale din interiorul și de pe frontiera pătratului 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 DFCE} ., cu 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 T_n} numărul punctelor laticeale din interiorul și de pe frontiera triunghiului 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 CAB} și cu 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_n} numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu 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 DBE} .
Avem 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_n = \left(2n^2+3n-1\right)^2,} 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 T_n = \sum\limits_{k=1}^{2n^2+n} k = \dfrac{1}{2}n\left(2n+1\right)\left(2n^2+n+1\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 S_n = \sum\limits_{k=2}^{2n+1} \left(2n^2+3n+1-g\left(k\right)\right) = \dfrac{1}{3}n\left(2n+1\right)\left(4n+1\right).} Atunci 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_n = A_n - 2S_n -T_n+3} , în formula precedenă de adaugă 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</math> pentru a corecta faptul că punctele <math>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} , respectiv 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 D} sunt puncte comune ale regiunilor 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 ADF} , 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 BDE} , respectiv 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 CAB} . Se obține 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_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.} Cazuri particulare: 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_1 = 3} este ușor de construit și verificat, 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_2 = 57} este reprezentat în figura de mai sus, 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_3 = 266 } 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 M_4 = 778} .