Tabara MathTime-Seniori,Ziua II-Cauchy-Schwarz-M.Lascu

Tabara MathTime-Seniori,Ziua II-Cauchy-Schwarz-M.Lascu

Mesajde Mr. Ady » Sâm Sep 03, 2011 9:23 pm

Curs sustinut de domn profesor Mircea Lascu

Fie a_{i}, b_{i} \in \mathbb{R}, atunci
(a^{2}_{1}+a^{2}_{2}+...+a^{2}_{n})(b^{2}_{1}+b^{2}_{2}+...+b^{2}_{n})\geq(a_{1}b_{1}+...+a_{n}b_{n})^{2}
Inegalitatea Cauchy-Schwarz

Problema 1. Demonstrati ca
a) * a^{3}+b^{3}\geq a^{2}+b^{2} daca a>0, b>0 si a^{2}+b^{2}\geq a+b
b) * a^{2}+b^{2}+c^{2}\geq 14 daca a+2b+3c\geq 14
c) * ab+\sqrt{(1-a^{2})(1-b^{2})\leq 1} daca |a|\leq 1, |b|\leq 1
d) ** a\sqrt{a^{2}+b^{2}}+c\sqrt{b^{2}+c^{2}}\leq a^{2}+b^{2}+c^{2}
e) ** \sqrt{a+b+c}\geq \sqrt{a-1}+\sqrt{b-1}+\sqrt{c-1} daca a,b,c>1 si \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=2

Problema 2. Demonstrati ca
a) * \frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\geq\frac{3}{2},\ a,b,c>0 - Nesbitt 1903

b) * \frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{a+d}+\frac{d}{a+b}\geq 2,\ a,b,c,d>0

c) ** \frac{a^{3}}{a^{2}+ab+b^{2}}+\frac{b^{3}}{b^{2}+bc+c^{2}}+\frac{c^{3}}{c^{2}+ca+a^{2}}\geq\frac{a+b+c}{3},\ a,b,c>0

d) *** \frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)}\geq\frac{3}{2}, unde a,b,c>0 si abc=1.

Problema 3. Demonstrati inegalitatile
a) ** (a_{1}+a_{2}+\ldots +a_{n})(a_{1}^{7}+a_{2}^{7}+\ldots +a_{n}^{7})\geq(a_{1}^{3}+a_{2}^{3}+\ldots +a_{n}^{3})(a_{1}^{5}+a_{2}^{5}+\ldots +a_{n}^{5}), unde a_{1},a_{2},\ldots,a_{n}>0.

b) ** (a_{1}^{k+1}+a_{2}^{k+1}+\ldots +a_{n}^{k+1} ) \right) \left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots +\frac{1}{a_{n}}\right) \geq n(a_{1}^{k}+a_{2}^{k}+\ldots +a_{n}^{k}), unde n,k\in\nn si a_{1},a_{2},\ldots, a_{n}>0.

c) ** \frac{a_{1}^{k}+a_{2}^{k}+\ldots +a_{n}^{k}}{n}\geq\left(\frac{a_{1}+a_{2}+\ldots a_{n}}{n}\right)^{k}, unde n,k\in\nn si a_{1},a_{2},\ldots,a_{n}>0.

Problema 4. Demonstrati inegalitatile

a) * \sqrt{a+1}+\sqrt{2a-3}+\sqrt{50-3a}\leq 12;

b) ** a+b+c\le abc+2, unde a^{2}+b^{2}+c^{2}=2;

c) ** 8(a^{3}+b^{3}+c^{3})^2\geq 9(a^{2}+bc)(b^{2}+ca)(c^{2}+ab),\ a,b,c>0;

d) ** 1\geq a_{1}a_{2}+a_{2}a_{3}+\ldots +a_{n-1}a_{n}+a_{n}a_{1}\geq -1, unde a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}=1;

e) *** a_{1}^{4}+a_{2}^{4}+\ldots +a_{n}^{4}\geq a^{3}_{1}a_{2}+a_{2}^{3}a_{3}+\ldots +a_{n-1}^{3}a_{n}+a^{3}_{n}a_{1}.

Problema 5. * Gasiti distanta de la punctul A(x_{0},y_{0}) la dreapta ax+by+c=0 (a^{2}+b^{2}\neq 0).

Problema 6. Gasiti cea mai mica valoare a expresiei:

a) * 2x+3y+4z, unde x^{2}+y^{2}+z^{2}=1;

b) ** (x-y)^{2}+\left(\sqrt{2-x^{2}}-\frac{9}{y}\right)^{2}, unde 0<x<\sqrt{2}, y>0.

Nota: Numarul de * reprezinta gradul de dificultate al problemei
* - usor, ** - mediu, *** - greu, **** - foarte greu
Solutiile COMPLETE vor fi trimise la linkurile problemelor respective
Catană Adrian,
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Mr. Ady
 
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Re: Tabara MathTime-Seniori,Ziua II-Cauchy-Schwarz-M.Lascu

Mesajde aecksteinul » Joi Sep 08, 2011 12:50 pm

La 1d) inegalitatea de demonstrat ar trebui sa fie

a\sqrt{a^2+b^2}+c\sqrt{c^2+b^2}\leq a^2+b^2+c^2

Iar la 4c) banuiesc ca trebuie aratat ca

8(a^3+b^3+c^3)^2\geq9(a^2+bc)(b^2+ca)(c^2+ab)
unde a,b,c>0.
aecksteinul
 
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Membru din: Dum Iun 26, 2011 9:01 pm


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