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10:18, 11 December 2024 AntalKrisztian talk contribs created page 16402 (Created page with "'''16402 (Cristina Vijdeluc, Salonic și Mihai Vijdeluc, Baia Mare)''' ''Fie <math>n \in \mathbb{N}^*</math> și numerele pozitive <math>x_1, x_2, \dots, x_n</math> care verifică relația <math>\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n} = \frac{n}{n+1}.</math> Arătați că <math>\frac{1}{x_1 + 2} + \frac{1}{x_2 + 6} + \dots + \frac{1}{x_n + n(n+1)} \leq \frac{n}{2(n+1)}.</math>'' '''Soluție:''' Din inegalitatea mediilor armonică și aritmetică,...")
10:08, 11 December 2024 Ghisa Catalin talk contribs created page 15324 (Created page with "'''E:15324 (Cristina Vijdeluc și Mihai Vijdeluc)''' ''Determinați cifrele nenule <math>x</math> și <math>y</math> pentru care <math>\frac{y^3}{x^3} - \frac{xy}{x} = 2\left(\frac{xy}{x} - 16\right)</math>.'' '''Soluție.''' Avem: <math>\frac{y^3}{x^3} - 3\frac{y}{x} + 32 = 0</math> și, cum <math>\frac{xy}{x} = \frac{10x + y}{x} = 10 + \frac{y}{x}</math>, obținem: <math>\frac{y^3}{x^3} - 3\frac{y}{x} + 2 = 0</math>. Notăm <math>\frac{y}{x} = t > 0</math> și obț...")
10:01, 11 December 2024 AntalKrisztian talk contribs created page 16393 (Created page with "'''16393 (Cristina Vijdeluc și Mihai Vijdeluc)''' ''Determinați numerele naturale nenule <math>n</math> pentru care numărul <math>a(n) = \sqrt{(1 \cdot 2 \cdot 3 \cdots n)^5 + 145}</math> este natural.'' '''Soluție:''' Calculând valorile lui <math>a(n)</math> pentru <math>n \in \{1, 2, 3, 4\},</math> găsim că doar pentru <math>n = 3,</math> <math>a(n) = 89</math> verifică cerința. Pentru <math>n \geq 5,</math> <math>(1 \cdot 2 \cdot 3 \cdots n)^5 + 145</ma...") Tag: Visual edit: Switched
09:57, 11 December 2024 Ghisa Catalin talk contribs created page 15323 (Created page with "<nowiki>'''</nowiki>E:15323 (Cristina Vijdeluc și Mihai Vijdeluc)<nowiki>'''</nowiki>") Tag: Visual edit
12:25, 10 December 2024 Cristina94 talk contribs created page S:E15313 (Created page with "'''S:E15313 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)''' ''Determinați cifra <math>a</math> pentru care fracția <math>\frac{a^2 \cdot \overline{aa} + a^2(a^2+1)}{\overline{202a}}</math> este echiunitară.'' '''Soluție:''' ''Fracția este echiunitară dacă <math>a^2 \cdot \overline{aa} + a^2(a^2+1) = \overline{202a}</math> sau <math>a^4 + 11a^3 + a^2 = \overline{202a}</math>. Avem <math> 2020 \leq \overline{202a} \leq 2029 </math>. Dacă <math> a \leq 4 <...")
12:10, 10 December 2024 Cristina94 talk contribs created page S:E15310 (Created page with "'''S:E15310 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)''' ''Arătați că nu există numere naturale <math>p</math> și <math>q</math> astfel încât să fie adevărată relația <math>p^2 - 2018 = 2^q</math>.'' '''Soluție:''' ''Putem scrie <math>p^2 = 2^q + 2018</math>. Pentru <math>q = 0</math> obținem <math>p^2 = 2019</math>, iar pentru <math>q = 1</math> obținem <math>p^2 = 2020</math> care nu sunt pătrate de numere naturale. Pentru <math>q \geq 2</ma...")
04:47, 10 December 2024 Andrei.Horvat talk contribs created page File:Gm 1-1977 e-5756.png
04:47, 10 December 2024 Andrei.Horvat talk contribs uploaded File:Gm 1-1977 e-5756.png
04:43, 10 December 2024 Andrei.Horvat talk contribs created page E:5756 (Created page with "'''E:5756 (Dumitru Acu)''' ''Fie <math>ABCD</math> un romb. Prin vârful <math>A</math> ducem o dreaptă arbitrară care intersectează pe <math>BC</math> în <math>E</math>, pe <math>DC</math> în <math>F</math>, iar pe diagonala <math>BD</math> în <math>G</math>. Să se arate că dreapta <math>CG</math> este tangentă în <math>C</math> cercului circumscris triunghiului <math>ECF</math>.''")
04:31, 8 December 2024 Andrei.Horvat talk contribs created page E:5743 (Created page with "'''E:5743 (Grigore Balog)''' ''Fie <math>A</math> mulțimea numerelor de forma <math>\overline{7a8b}</math>, care se divid cu <math>15</math> și <math>B</math> mulțimea numerelor de forma <math>\overline{7x8y}</math>, care se divid cu <math>40</math>. Să se determine mulțimile <math>A \cup B</math>, <math>A\cap B</math> și <math>A\setmnus B</math>.''")
04:04, 8 December 2024 Andrei.Horvat talk contribs created page E:5763 (Created page with "'''E:5763 (Tudor Rițiu)''' ''Un lot în formă de trapez isoscel are, în metri, baza mică egală cu valoarea numerică a expresiei''<math display="block">E\left(x,y\right) = \frac{\left(x+y\right)^2 - x-y}{x^2-y^2}:\left[ 1 - \frac{1}{x+y}\right]</math>''pentru'' <math>x</math> ''a cărei valoare satisface proporția'' <math display="block">\frac{x-2,5}{1\frac{1}{2}} = \frac{x}{2}</math> ''iar'' <math>y</math> ''fiind a treia parte din'' <math>x</math>. '' Latura nep...")
03:55, 8 December 2024 Andrei.Horvat talk contribs created page Gazeta matematică 1977 (Created page with "== Gazeta Matematică 1/1977 == '''E:5763 (Tudor Rițiu)''' ''Un lot în formă de trapez isoscel are, în metri, baza mică egală cu valoarea numerică a expresiei''")
13:01, 5 December 2024 Andrei.Horvat talk contribs created page S:E22.136 (Created page with "'''S:E22.136 (Cristina Vijdeluc, Mihai Vijdeluc)''' '' Aflați numerele <math>n</math>, <math>p</math> natuarale neule pentru care <math>2n^2 = 253 \cdot \left(p! + 2022)</math>, unde <math>p! = 1\cdot 2\cdot 3\cdot \ldots\cdot p</math>.")
13:58, 3 December 2024 Hotico Iulia Denisa talk contribs created page S:E15.316 (Created page with "'''S:E15316 (Cristina Vijdeluc și Mihai Vijdeluc)''' ''Determinați numărul prim <math>p</math> și numărul natural <math>q</math> astfel încât'' ''<math display="block">p^{2} + 5^{p} + 31 = 3181^{q}</math>.'' '''Soluție:''' ''Putem scrie <math>p^{2} + 5^{p} = 3181^{q} - 31</math>. Cum ultima cifră a lui <math>3181^{q} - 31</math> este <math>0</math> și <math>5^{p}</math> este divizibil cu <math>5</math>, deducem că <math>p=5</math>. Atunci relația devine <...")
13:44, 3 December 2024 Hotico Iulia Denisa talk contribs created page S:E15.314 (Created page with "'''S:E15.314 (Cristina Vijdeluc și Mihai Vijdeluc)''' ''Determinați numerele naturale <math>m</math> și <math>n</math> pentru care este adevărată relația <math>2018^{m}=8^{n}+2010</math>.'' '''Soluție:''' ''Pentru <math>n=0</math> și <math>m=0</math> nu avem soluție. Pentru <math>n=1</math> obținem <math>m=1</math>. Dacă scriem relația sub forma <math>2018^{m}-2010=8^{n}</math>, atunci pentru <math>n>1</math> și <math>m>1</math> avem în dreapta un număr...")
08:37, 1 December 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2017 (Created page with "== Gazeta Matematică 6-7-8/2017 == '''27401 (Radu Pop)''' ''Fie <math>n \in \mathbb{N}</math>. Să se arate că <math display="block"> (n+1)ab(a+b)-(4n^3+13n+10)ab+(n+2)^3(a+b) \geq (n+2)^3, </math> oricare ar fi <math>a,b \in [1,\infty)</math>''")
08:21, 1 December 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2015 (Created page with "== Gazeta Matematică 1/2015 == == Gazeta Matematică 2/2015 == == Gazeta Matematică 3/2015 == == Gazeta Matematică 9/2015 ==")
06:58, 1 December 2024 Heres.Gabriela talk contribs created page S:E15.325 (Created page with "'''S:E15.325 (Cristina Vijdeluc și Mihai Vijdeluc, Baia Mare)''' ''Determinațti numerele naturale \( \overline{abcd} \) pentru care ..'' '''Soluție:''' ''''")
17:50, 30 November 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2021 (Created page with "== Gazeta Matematică 6-7-8/2021 == == Gazeta Matematică 10/2021 == '''28163 (Dana Heuberger)''' ''Aflați șirul de numere naturale nenule <math>(a_n)_{n\geq1}</math> pentru care <math>\frac{1}{{(1+a_1) \cdot a_{a_1}}} + \frac{1}{{(1+a_2) \cdot a_{a_2}}} + \ldots + \frac{1}{{(1+a_n) \cdot a_{a_n}}} = \frac{n}{{n+1}}</math>, pentru orice <math>n \geq 1</math>.'' == Gazeta Matematică 11/2021 == == Gazeta Matematică 12/2021 ==")
16:52, 30 November 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2014 (Created page with "== Gazeta Matematică 1/2013 == == Gazeta Matematică 11/2013 == '''E:14742 (Liliana Puț)''' ''a) Arătați că oricare ar fi numerele reale <math>a</math>, <math>b</math>, <math>c</math> avem''<math display="block">|a + b| + |a + c| \ge |b - c|.</math>''b) Demonstrați că pentru orice număr real <math>x</math> avem''<math display="block">|x + 1| + |x + 2| + |x + 3| + ... + |x + 2014| \ge 1007^2.</math>")
16:47, 30 November 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2013 (Created page with "== Gazeta Matematică 1/2013 == '''26713 (Radu Pop și Vasile Ienuțaș)''' ''Se consideră șirul de numere reale <math>(x_n)_{n \geq 0}</math> și <math>(y_n)_{n \geq 0}</math> cu <math>x_n \geq 1</math>, <math>y_n \geq 1</math>, pentru orice <math>n \in \mathbb{N}</math>, și <math>\lim_{{n \to \infty}} (x_n^2 + y_n^2) = 2</math>. Să se calculeze <math>\lim_{{n \to \infty}} x_n</math> și <math>\lim_{{n \to \infty}} y_n</math>.''")
16:11, 30 November 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2012 (Created page with "== Gazeta Matematică 9/2012 == '''E:14383 (Gheorghe Gherasim)''' ''Numerele naturale distincte'' <math>a</math>'','' <math>b</math> ''verifică <math>9 \cdot [\,a, b]\,=a \cdot b \cdot (\,a \cdot b)\,</math>.'' i) ''Arătați că'' <math>a</math> ''și'' <math>b</math> ''nu sunt prime între ele.'' ii) ''Arătați că diferența numerelor este cel puțin'' <math>3</math>''.'' ''Se consideră că'' <math>[a,b]</math> ''reprezintă cel mai mic multiplu comun al numer...")
15:59, 30 November 2024 Andrei.Horvat talk contribs created page Gazeta matematică 2011 (Created page with "== Gazeta Matematică 7-8-9/2011 == '''E:14228 (Mihai Vijdeluc)''' ''Arătați că nu există niciun număr de forma <math>\overline{abc}</math> cu proprietatea că <math>\sqrt{\overline{abc}} = \sqrt{\overline{bc}} + a</math>''")
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