Syv ulikheter fra syd
Lagt inn: 03/04-2014 19:09
1) La $x,y,z$ være positive reelle tall slik at $xyz=1$. Vis at $\frac1{x+y^{20}+z^{11}}+\frac1{y+z^{20}+x^{11}}+\frac1{z+x^{20}+y^{11}}\leq 11$
2) La $a,b,c$ være positive reelle tall slik at $a+b+c=1$. Vis at $\frac{a^2b^2}{c^3(a^2-ab+b^2)}+\frac{b^2c^2}{a^3(b^2-bc+c^2)}+\frac{c^2a^2}{b^3(c^2-ca+a^2)}\geq \frac{3}{ab+bc+ca}$
3) For positive reelle tall $a,b,c,d$, vis at $\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4}\geq 2\sqrt{2}(ad+bc)$
4) La $a,b,c$ være reelle tall slik at $0\leq a\leq b\leq c$. Vis at $(a+3b)(b+4c)(c+2a)\geq 60abc$
5) La $x,y,z$ være positive reelle tall slik at $xy+yz+zx=1$. Vis at $\frac{27}{4}(x+y)(y+z)(z+x)\geq (\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x})^2\geq 6\sqrt{3}$
6) La $a,b,c$ være postive reelle tall slik at $a+b+c=1$. Vis at $\frac{1}{ab+2c^2+2c}+\frac{1}{bc+2a^2+2a}+\frac{1}{ca+2b^2+2b}\geq \frac{1}{ab+bc+ca}$
7) La $a,b,c$ være positive reelle tall slik at $ab+bc+ca\leq 1$. Vis at $a+b+c+\sqrt 3\geq 8abc\left(\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2}\right)$
2) La $a,b,c$ være positive reelle tall slik at $a+b+c=1$. Vis at $\frac{a^2b^2}{c^3(a^2-ab+b^2)}+\frac{b^2c^2}{a^3(b^2-bc+c^2)}+\frac{c^2a^2}{b^3(c^2-ca+a^2)}\geq \frac{3}{ab+bc+ca}$
3) For positive reelle tall $a,b,c,d$, vis at $\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4}\geq 2\sqrt{2}(ad+bc)$
4) La $a,b,c$ være reelle tall slik at $0\leq a\leq b\leq c$. Vis at $(a+3b)(b+4c)(c+2a)\geq 60abc$
5) La $x,y,z$ være positive reelle tall slik at $xy+yz+zx=1$. Vis at $\frac{27}{4}(x+y)(y+z)(z+x)\geq (\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x})^2\geq 6\sqrt{3}$
6) La $a,b,c$ være postive reelle tall slik at $a+b+c=1$. Vis at $\frac{1}{ab+2c^2+2c}+\frac{1}{bc+2a^2+2a}+\frac{1}{ca+2b^2+2b}\geq \frac{1}{ab+bc+ca}$
7) La $a,b,c$ være positive reelle tall slik at $ab+bc+ca\leq 1$. Vis at $a+b+c+\sqrt 3\geq 8abc\left(\frac1{1+a^2}+\frac1{1+b^2}+\frac1{1+c^2}\right)$