Gọi 46 số tự nhiên liên tiếp là

\(n;n+1;n+2;n+3;...;n+45\) với \(n\in N\)

Ta có

\(n+\left(n+1\right)+\left(n+2\right)+\left(n+3\right)+...+\left(n+45\right)\)

\(=46n+\left(1+2+3+...+45\right)\)

\(=46n+\dfrac{45.46}{2}=46n+1035\)

Vì \(1035⋮̸46\) nên \(\left(46n+1035\right)⋮̸46\)

Hay tổng của 46 số tự nhiên liên tiếp ko chia hết cho 46

1. \(\left(1+\dfrac{a+\sqrt{a}}{\sqrt{a}+1}\right)\left(1-\dfrac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)

\(=\left(1+\dfrac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right)\left(1-\dfrac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)\)

\(=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)

2. a) Với a>b>0 thì

\(Q=\dfrac{a}{\sqrt{a^2-b^2}}-\left(1+\dfrac{a}{\sqrt{a^2-b^2}}\right):\dfrac{b}{a-\sqrt{a^2-b^2}}\)

\(=\dfrac{a}{\sqrt{a^2-b^2}}-\dfrac{a+\sqrt{a^2-b^2}}{\sqrt{a^2-b^2}}.\dfrac{a-\sqrt{a^2-b^2}}{b}\)

\(=\dfrac{a}{\sqrt{a^2-b^2}}-\dfrac{a^2-\left(a^2-b^2\right)}{b\sqrt{a^2-b^2}}\)

\(=\dfrac{a}{\sqrt{a^2-b^2}}-\dfrac{b^2}{b\sqrt{a^2-b^2}}=\dfrac{a}{\sqrt{a^2-b^2}}-\dfrac{b}{\sqrt{a^2-b^2}}\)

\(=\dfrac{a-b}{\sqrt{a^2-b^2}}=\dfrac{a-b}{\sqrt{a-b}.\sqrt{a+b}}=\sqrt{\dfrac{a-b}{a+b}}\)

b) Thay a = 3b ta được

\(Q=\sqrt{\dfrac{a-b}{a+b}}=\sqrt{\dfrac{3b-b}{3b+b}}=\sqrt{\dfrac{2b}{4b}}=\sqrt{\dfrac{1}{2}}=\dfrac{\sqrt{2}}{2}\)

Vì a,b,c>0 áp dụng BĐT Cauchy ta có

\(\sqrt{\dfrac{a}{b+c}}=\dfrac{a}{\sqrt{a\left(b+c\right)}}\ge\dfrac{a}{\dfrac{a+b+c}{2}}=\dfrac{2a}{a+b+c}\) (1)

Tương tự \(\sqrt{\dfrac{b}{c+a}}\ge\dfrac{2b}{a+b+c}\) (2) và \(\sqrt{\dfrac{c}{a+b}}\ge\dfrac{2c}{a+b+c}\) (3)

Cộng (1), (2), (3) vế theo vế

\(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}\)

\(\ge2\left(\dfrac{a}{a+b+c}+\dfrac{b}{a+b+c}+\dfrac{c}{a+b+c}\right)=2\)

Đẳng thức xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}a=b+c\\b=c+a\\a=b+c\end{matrix}\right.\) \(\Leftrightarrow a=b=c=0\) (vô lý)

Vậy đẳng thức ko xảy ra, hay \(\sqrt{\dfrac{a}{b+c}}+\sqrt{\dfrac{b}{c+a}}+\sqrt{\dfrac{c}{a+b}}>2\)

Ta có \(x+y+z\ge\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\)

\(\Leftrightarrow\left(x-2\sqrt{xy}+y\right)+\left(y-2\sqrt{yz}+z\right)+\left(z-2\sqrt{zx}+x\right)\ge0\)

\(\Leftrightarrow\left(\sqrt{x}-\sqrt{y}\right)^2+\left(\sqrt{y}-\sqrt{z}\right)^2+\left(\sqrt{z}-\sqrt{x}\right)^2\ge0\) (luôn đúng)

Vậy \(x+y+z\ge\sqrt{xy}+\sqrt{yz}+\sqrt{zx}=1\)

Theo BĐT Cauchy-Schwarz dạng Engel

\(A=\dfrac{x^2}{x+y}+\dfrac{y^2}{y+z}+\dfrac{z^2}{z+x}\ge\dfrac{\left(x+y+z\right)^2}{2\left(x+y+z\right)}=\dfrac{x+y+z}{2}\ge\dfrac{1}{2}\)

Đẳng thức xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\dfrac{x}{x+y}=\dfrac{y}{y+z}=\dfrac{z}{z+x}\\\sqrt{xy}+\sqrt{yz}+\sqrt{zx}=1\end{matrix}\right.\)

\(\Leftrightarrow a=b=c=\dfrac{1}{3}\)

có dạng toán giống bài bạn đăng đấy!

tương tự mà làm.

\(x^3-1=0\)

\(\Leftrightarrow x^3=1\)

\(\Leftrightarrow x=1\)