$y=\left(\mathrm{m\; -\; 1}\right)x^2\mathrm{+2mx\; -3m\; +\; 1}\left(P_m\right)$ \sqrt{9x^2+3}+9x-1\ge \sqrt{9x^2+15}$$ f\left(x\right)=a_0+a_1.cosx+a_2.cos2x+...+a_n.cosnx$$ f\left(x\right)>0\forall x\in R$$ a_0>0$x^2+(x+1{)}^2=y^4+(y+1{)}^4$ a,b,c$$ abc=1$$ \frac{ab}{a^5+b^5+ab}+\frac{bc}{b^5+c^5+bc}+\frac{ca}{c^5+a^5+ca}\le 1$$ a,b,c$$ a^2+b^2+c^2=1$$ \frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}$$ a\ge 6$$ a^2+\frac{6}{\sqrt{a}}-\sqrt{6}\ge 36$$ a,b,c,d$$ 1\le a\le b\le c\le d\le 90$$ P=\frac{a}{b}+\frac{3c}{d}$$ x,a,b,c$$ x^2=a^2+b^2+c^2$$ \frac{a}{x+2a}+\frac{b}{x+2b}+\frac{c}{2+2c}\le \frac{3}{2+\sqrt{3}}$$ y=2+\sqrt{2}\sin \left(x+\frac{\Pi }{4}\right)+2\sqrt{1+\sin x+\cos x+\sin x\cos x}$$ x\in R$$ x>0$$ y>0$$ x+2y<\frac{5\Pi }{4}$$ \cos \left(x+y\right)<\frac{y\sin x}{x\sin y}$$ y=\frac{2x-1}{x+1}$\frac{5}{2\times 1}+\frac{4}{1\times 11}+\frac{3}{11\times 2}+\frac{1}{2\times 15}+\frac{13}{15\times 4}+\frac{15}{4\times 43}+\frac{13}{43\times 8}y=5+2\sqrt{-x^2+4x-3}-4\sqrt{x-1}-4\sqrt{3-x}\{\begin{array}{c}{x}^2+y+x^{3}y+x{y}^2+xy=\frac{-5}{4}\\ x^4+y^2+xy\left(1+2x\right)=\frac{-5}{4}\end{array}\left(x_1;y_1\right);\left(x_2;y_2\right)x_1^3+x_2^3$ +\mathrm{2\backslash (\backslash overrightarrow\{MB\}\backslash )}+3\backslash (\backslash overrightarrow\{MC\}\backslash )$$ +\mathrm{2\backslash (\backslash overrightarrow\{MB\}\backslash )}-\mathrm{4\backslash (\backslash overrightarrow\{MC\}\backslash )}$si{n}^{2}A+si{n}^{2}B+co{s}^{2}C+\frac{1}{4}=2sinA.sinB+cosC.$