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28868 - Revision history
2026-05-01T05:58:24Z
Revision history for this page on the wiki
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https://wiki.universitas.ro/index.php?title=28868&diff=10664&oldid=prev
Andrei.Horvat at 08:27, 4 August 2025
2025-08-04T08:27:40Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:27, 4 August 2025</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''[[28868]] (<del style="font-weight: bold; text-decoration: none;">Andre </del>Horvat-Marc)'''</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''[[28868]] (<ins style="font-weight: bold; text-decoration: none;">Andrei </ins>Horvat-Marc)'''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n\in \mathbb{N^\ast}</math> și funcțiile <math>f:\left[0,2n^2+3n\right] \to \left[1,2n+1\right]</math>, <math> f\left(x\right) = \frac{\sqrt{8x+9}-1}{2}</math> ''și ''<math> g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math>, <math> g\left(x\right) = f^{-1}\left(x\right)</math>.''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n\in \mathbb{N^\ast}</math> și funcțiile <math>f:\left[0,2n^2+3n\right] \to \left[1,2n+1\right]</math>, <math> f\left(x\right) = \frac{\sqrt{8x+9}-1}{2}</math> ''și ''<math> g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math>, <math> g\left(x\right) = f^{-1}\left(x\right)</math>.''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10663&oldid=prev
Andrei.Horvat at 08:26, 4 August 2025
2025-08-04T08:26:27Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:26, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l26">Line 26:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Se obține</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Se obține</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), n\in \mathbb{N}^\ast.</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), n\in \mathbb{N}^\ast.</math></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Laticeale-5-2024.png|thumb|304x304px]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Cazuri particulare:''' </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Cazuri particulare:''' </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_1 = 3</math> este ușor de construit și verificat, </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_1 = 3</math> este ușor de construit și verificat, </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[File:Laticeale-5-2024.png|thumb|304x304px]]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_2 = 57</math> este reprezentat în figura alăturată, </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_2 = 57</math> este reprezentat în figura alăturată, </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10662&oldid=prev
Andrei.Horvat at 08:25, 4 August 2025
2025-08-04T08:25:26Z
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<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:25, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l36">Line 36:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Observație'''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Observație'''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Problema de mai sus este echivalentă cu următoarea problemă</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Problema de mai sus este echivalentă cu următoarea problemă</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n\in \mathbb{N}^\ast</math> un număr natural și funcția <math>f:\left[0,2n^2+3n\right]\to \left[1,2n+1\right]</math>, cu <math>f\left(x\right) = \dfrac{\sqrt{8x+9}-1}{2}</math>. Determinați valoarea sumei <math> S\left(n\right)=\sum\limits_{k=0}^{2n^2+3n} \left[f\left(k\right)\right]</math>, unde prin <math>\left[a\right]</math> s-a notat partea întreagă a numărului real <math>a\in \mathbb{R}</math>.''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie <math>n\in \mathbb{N}^\ast</math> un număr natural și funcția <math>f:\left[0,2n^2+3n\right]\to \left[1,2n+1\right]</math>, cu <math>f\left(x\right) = \dfrac{\sqrt{8x+9}-1}{2}</math>. Determinați valoarea sumei <math> S\left(n\right)=\sum\limits_{k=0}^{2n^2+3n} \left[f\left(k\right)\right]</math>, unde prin <math>\left[a\right]</math> s-a notat partea întreagă a numărului real <math>a\in \mathbb{R}</math>.''</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10661&oldid=prev
Andrei.Horvat at 08:24, 4 August 2025
2025-08-04T08:24:42Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:24, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l35">Line 35:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Observație'''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''Observație'''</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Problema de mai sus este echivalentă cu următoarea problemă</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">''Fie <math>n\in \mathbb{N}^\ast</math> un număr natural și funcția <math>f:\left[0,2n^2+3n\right]\to \left[1,2n+1\right]</math>, cu <math>f\left(x\right) = \dfrac{\sqrt{8x+9}-1}{2}</math>. Determinați valoarea sumei <math> S\left(n\right)=\sum\limits_{k=0}^{2n^2+3n} \left[f\left(k\right)\right]</math>, unde prin <math>\left[a\right]</math> s-a notat partea întreagă a numărului real <math>a\in \mathbb{R}</math>.''</ins></div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10660&oldid=prev
Andrei.Horvat at 08:21, 4 August 2025
2025-08-04T08:21:57Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:21, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22">Line 22:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> 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><del style="font-weight: bold; text-decoration: none;">, </del>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</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 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 <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>M_n = A_n - 2S_n -T_n+3<ins style="font-weight: bold; text-decoration: none;">,</ins></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>. </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Se obține</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), n\in \mathbb{N}^\ast.</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math display="block">M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), n\in \mathbb{N}^\ast.</math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Cazuri particulare: <math>M_1 = 3</math> este ușor de construit și verificat, <math>M_2 = 57</math> este reprezentat în figura <del style="font-weight: bold; text-decoration: none;">de mai sus</del>, <math>M_3 = 266 </math> și <math>M_4 = 778</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''</ins>Cazuri particulare:<ins style="font-weight: bold; text-decoration: none;">''' </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>M_1 = 3</math> este ușor de construit și verificat, </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Laticeale-5-2024.png|thumb|304x304px]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>M_2 = 57</math> este reprezentat în figura <ins style="font-weight: bold; text-decoration: none;">alăturată</ins>, </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math>M_3 = 266 </math> și <math>M_4 = 778</math>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Observație'''</ins></div></td></tr>
</table>
Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10658&oldid=prev
Andrei.Horvat at 08:15, 4 August 2025
2025-08-04T08:15:09Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:15, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23">Line 23:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 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</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 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</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), <del style="font-weight: bold; text-decoration: none;">\enskip </del>n\in \mathbb{N}^\ast.</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><math <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), n\in \mathbb{N}^\ast.</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
</table>
Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10657&oldid=prev
Andrei.Horvat at 08:14, 4 August 2025
2025-08-04T08:14:10Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:14, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l22">Line 22:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> 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<del style="font-weight: bold; text-decoration: none;">&lt;</del>/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</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 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<ins style="font-weight: bold; text-decoration: none;"><</ins>/math> pentru a corecta faptul că punctele <math>A</math>, <math>B</math>, respectiv <math>D</math><ins style="font-weight: bold; text-decoration: none;">, </ins>sunt puncte comune ale regiunilor <math>ADF</math>, <math>BDE</math>, respectiv <math>CAB</math>. Se obține</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
</table>
Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10656&oldid=prev
Andrei.Horvat at 08:07, 4 August 2025
2025-08-04T08:07:20Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:07, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> 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></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 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. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> S_n</del></math> numărul punctelor laticeale din interiorul și de pe frontiera <del style="font-weight: bold; text-decoration: none;">triunghiului curbiliniu </del><math><del style="font-weight: bold; text-decoration: none;">DBE</del></math>, <del style="font-weight: bold; text-decoration: none;">respectiv <math>DAF<math>. Datorită simetriei triunghiurile curbilinii <math>DBE<math> și <math>DAF</math> \\ </del><math>T_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului <math>CAB</math><del style="font-weight: bold; text-decoration: none;">\\ </del><math><del style="font-weight: bold; text-decoration: none;">A_n</del></math> numărul punctelor laticeale din interiorul și de pe frontiera <del style="font-weight: bold; text-decoration: none;">pătratului </del><math><del style="font-weight: bold; text-decoration: none;">DFCE</del></math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> </ins>Notăm <ins style="font-weight: bold; text-decoration: none;">cu </ins><math><ins style="font-weight: bold; text-decoration: none;">A_n</ins></math> numărul punctelor laticeale din interiorul și de pe frontiera <ins style="font-weight: bold; text-decoration: none;">pătratului </ins><math><ins style="font-weight: bold; text-decoration: none;">DFCE</ins></math><ins style="font-weight: bold; text-decoration: none;">.</ins>, <ins style="font-weight: bold; text-decoration: none;">cu </ins><math>T_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului <math>CAB</math> <ins style="font-weight: bold; text-decoration: none;">și cu </ins><math></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> S_n</ins></math> numărul punctelor laticeale din interiorul și de pe frontiera <ins style="font-weight: bold; text-decoration: none;">triunghiului curbiliniu </ins><math><ins style="font-weight: bold; text-decoration: none;">DBE</ins></math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> Avem <math>A_n = \left(2n^2+3n-1\right)^2</math><del style="font-weight: bold; text-decoration: none;">, </del><math>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>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></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> Avem <math <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>A_n = \left(2n^2+3n-1\right)^2<ins style="font-weight: bold; text-decoration: none;">,</ins></math> <math <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>T_n = \sum\limits_{k=1}^{2n^2+n} k = \dfrac{1}{2}n\left(2n+1\right)\left(2n^2+n+1\right)<ins style="font-weight: bold; text-decoration: none;">, </ins></math> și <math <ins style="font-weight: bold; text-decoration: none;">display="block"</ins>>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<ins style="font-weight: bold; text-decoration: none;">&lt;/</ins>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</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Atunci <math>M_n = A_n - 2S_n -T_n+3</math>, în formula precedenă de adaugă <math>3<del style="font-weight: bold; text-decoration: none;"><</del>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</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math>M_n = \dfrac{1}{6}\left(12n^4+28n^3-3n^2-43n+24\right), \enskip n\in \mathbb{N}^\ast.</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10655&oldid=prev
Andrei.Horvat at 08:03, 4 August 2025
2025-08-04T08:03:27Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:03, 4 August 2025</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Cum <math>g = f^{-1}</math>, se obține că funcția <math>g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math> este definită prin <math>g\left(x\right) = \dfrac{\left(x-1\right)\left(x+2\right)}{2}</math>. Mai mult, <math>g\left(k\right) \in \mathbb{N}</math> oricare ar fi <math>k \in \left[1, 2n+1\right] \cap \mathbb {N}</math>. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Cum <math>g = f^{-1}</math>, se obține că funcția <math>g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math> este definită prin <math>g\left(x\right) = \dfrac{\left(x-1\right)\left(x+2\right)}{2}</math>. Mai mult, <math>g\left(k\right) \in \mathbb{N}</math> oricare ar fi <math>k \in \left[1, 2n+1\right] \cap \mathbb {N}</math>. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Avem <math>B\left(2n+1,2n^2+3n\right) \in <del style="font-weight: bold; text-decoration: none;">G_f</del></math>, <math>A\left(2n^2+3n,2n+1\right) \in <del style="font-weight: bold; text-decoration: none;">G_g </del></math> și <math>G_f \cap G_g = \left\{D\left(2,2\right)\right\}<del style="font-weight: bold; text-decoration: none;"></math>. </del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Avem <math>B\left(2n+1,2n^2+3n\right) \in <ins style="font-weight: bold; text-decoration: none;">G_g</ins></math>, <math>A\left(2n^2+3n,2n+1\right) \in <ins style="font-weight: bold; text-decoration: none;">G_f </ins></math> și <math>G_f \cap G_g = \left\{D\left(2,2\right)\right\}</math>. </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> Au loc inegalitățile <math>f\left(x\right) \le x</math> oricare ar fi <math>x \in \left[2,2n^2+3n\right]</math> și <math>g\left(x\right) \ge x</math> oricare ar fi <math>x \in \left[2, 2n+1\right]</del></math>.</div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </del>Considerăm că mulțimea <math>M</math> este mulțimea tuturor punctelor din plan cuprinse în interiorul triunghiului curbiliniu <math>ABD</math>, deci este necesar să numărăm punctele laticeale din interiorul triunghiului curbiliniu <math>ABD</math>, vom nota cu <math>M_n</math> acest număr.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Au loc inegalitățile <math>f\left(x\right) \le x</math> oricare ar fi <math>x \in \left[2,2n^2+3n\right]</math> și <math>g\left(x\right) \ge x</math> oricare ar fi <math>x \in \left[2, 2n+1\right]</math>.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Considerăm că mulțimea <math>M</math> este mulțimea tuturor punctelor din plan cuprinse în interiorul triunghiului curbiliniu <math>ABD</math>, deci este necesar să numărăm punctele laticeale din interiorul triunghiului curbiliniu <math>ABD</math>, vom nota cu <math>M_n</math> acest număr.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> 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> și <del style="font-weight: bold; text-decoration: none;"> \\ %</del><math><del style="font-weight: bold; text-decoration: none;">AB \cap CD = \left\{N\right\}</del><math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> 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)<ins style="font-weight: bold; text-decoration: none;"></math>. Observăm că, datorită simetriei, triunghiurile curbilinii <math>DBE</ins></math> și <math><ins style="font-weight: bold; text-decoration: none;">DAF</ins><<ins style="font-weight: bold; text-decoration: none;">/</ins>math> <ins style="font-weight: bold; text-decoration: none;">conțin același număr de puncte laticeale</ins>. <ins style="font-weight: bold; text-decoration: none;">Notăm </ins><math></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </del><math>S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>, respectiv <math>DAF<math>. Datorită simetriei triunghiurile curbilinii <math>DBE<math> și <math>DAF</math> <del style="font-weight: bold; text-decoration: none;">conțin același număr de puncte laticeale.</del>\\ <math>T_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului <math>CAB</math>\\ <math>A_n</math> numărul punctelor laticeale din interiorul și de pe frontiera pătratului <math>DFCE</math>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> </ins>S_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului curbiliniu <math>DBE</math>, respectiv <math>DAF<math>. Datorită simetriei triunghiurile curbilinii <math>DBE<math> și <math>DAF</math> \\ <math>T_n</math> numărul punctelor laticeale din interiorul și de pe frontiera triunghiului <math>CAB</math>\\ <math>A_n</math> numărul punctelor laticeale din interiorul și de pe frontiera pătratului <math>DFCE</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Avem <math>A_n = \left(2n^2+3n-1\right)^2</math>, <math>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>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></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Avem <math>A_n = \left(2n^2+3n-1\right)^2</math>, <math>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>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></div></td></tr>
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Andrei.Horvat
https://wiki.universitas.ro/index.php?title=28868&diff=10654&oldid=prev
Andrei.Horvat at 07:56, 4 August 2025
2025-08-04T07:56:32Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:56, 4 August 2025</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie punctele'' <math>A\left(2n^2+3n,2n+1\right)</math>, <math>B\left(2n+1,2n^2+3n\right)</math> ''și mulțimea <math>M</math> a punctelor din plan cuprinse între graficele funcțiilor <math>f</math> și <math>g</math> și dreapta <math>AB</math>. Aflați numărul punctelor din <math>M</math> care au ambele coordonate întregi.''</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''Fie punctele'' <math>A\left(2n^2+3n,2n+1\right)</math>, <math>B\left(2n+1,2n^2+3n\right)</math> ''și mulțimea <math>M</math> a punctelor din plan cuprinse între graficele funcțiilor <math>f</math> și <math>g</math> și dreapta <math>AB</math>. Aflați numărul punctelor din <math>M</math> care au ambele coordonate întregi.''</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Soluție'''</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Soluție<ins style="font-weight: bold; text-decoration: none;">.</ins>''' </div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Cum <math>g = f^{-1}</math>, se obține că funcția <math>g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math> este definită prin <math>g\left(x\right) = \dfrac{\left(x-1\right)\left(x+2\right)}{2}</math>. Avem <math>B\left(2n+1,2n^2+3n\right) \in G_f</math>, <math>A\left(2n^2+3n,2n+1\right) \in G_g </math> și <math>G_f \cap G_g = \left\{D\left(2,2\right)\right\}</math>. </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Cum <math>g = f^{-1}</math>, se obține că funcția <math>g:\left[1,2n+1\right] \to \left[0,2n^2+3n\right]</math> este definită prin <math>g\left(x\right) = \dfrac{\left(x-1\right)\left(x+2\right)}{2}</math>. <ins style="font-weight: bold; text-decoration: none;">Mai mult, <math>g\left(k\right) \in \mathbb{N}</math> oricare ar fi <math>k \in \left[1, 2n+1\right] \cap \mathbb {N}</math>. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Avem <math>B\left(2n+1,2n^2+3n\right) \in G_f</math>, <math>A\left(2n^2+3n,2n+1\right) \in G_g </math> și <math>G_f \cap G_g = \left\{D\left(2,2\right)\right\}</math>. </div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Au loc inegalitățile <math>f\left(x\right) \le x</math> oricare ar fi <math>x \in \left[2,2n^2+3n\right]</math> și <math>g\left(x\right) \ge x</math> oricare ar fi <math>x \in \left[2, 2n+1\right]</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> Au loc inegalitățile <math>f\left(x\right) \le x</math> oricare ar fi <math>x \in \left[2,2n^2+3n\right]</math> și <math>g\left(x\right) \ge x</math> oricare ar fi <math>x \in \left[2, 2n+1\right]</math>.</div></td></tr>
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Andrei.Horvat