Vì \(ab+bc+ca=2020\)

\(\Rightarrow a^2+2020=a^2+ab+bc+ca\)

\(=\left(a^2+ab\right)+\left(bc+ca\right)=a\left(a+b\right)+c\left(a+b\right)\)

\(=\left(a+b\right)\left(a+c\right)\)

Tương tự ta có: \(b^2+2020=\left(b+a\right)\left(b+c\right)\)

                          \(c^2+2020=\left(c+b\right)\left(c+a\right)\)

\(\Rightarrow\frac{a^2-bc}{a^2+2020}+\frac{b^2-ca}{b^2+2020}+\frac{c^2-ab}{c^2+2020}\)

\(=\frac{a^2-bc}{\left(a+b\right)\left(a+c\right)}+\frac{b^2-ca}{\left(b+a\right)\left(b+c\right)}+\frac{c^2-ab}{\left(c+a\right)\left(c+b\right)}\)

\(=\frac{\left(a^2-bc\right)\left(b+c\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}+\frac{\left(b^2-ca\right)\left(c+a\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}+\frac{\left(c^2-ab\right)\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(=\frac{\left(a^2-bc\right)\left(b+c\right)+\left(b^2-ca\right)\left(c+a\right)+\left(c^2-ab\right)\left(a+b\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(=\frac{\left(a^2b+a^2c-b^2c-bc^2\right)+\left(b^2c+b^2a-c^2a-ca^2\right)+\left(c^2a+c^2b-a^2b-ab^2\right)}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(=\frac{a^2b+a^2c-b^2c-bc^2+b^2c+b^2a-c^2a-ca^2+c^2a+c^2b-a^2b-ab^2}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}\)

\(=\frac{0}{\left(a+b\right)\left(b+c\right)\left(c+a\right)}=0\)( đpcm )

Ta có 

  \(a^2+ab+bc+ca=\left(a+b\right)\left(a+c\right)\)

\(b^2+ab+bc+ac=\left(a+b\right)\left(b+c\right)\)

\(c^2+ab+bc+ac=\left(a+c\right)\left(b+c\right)\)

   Thay ab + bc + ac = 2020 vào biểu thức \(\frac{a^2-bc}{a^2+2020}+\frac{b^2-ca}{b^2+2020}+\frac{c^2-ab}{c^2+2020}\)có

       \(\frac{a^2-bc}{a^2+2020}+\frac{b^2-ca}{b^2+2020}+\frac{c^2-ab}{c^2+2020}\)

\(=\frac{a^2-bc}{\left(a+b\right)\left(a+c\right)}+\frac{b^2-ca}{\left(a+b\right)\left(b+c\right)}+\frac{c^2-ab}{\left(b+c\right)\left(a+c\right)}\)

\(=\frac{\left(a^2-bc\right)\left(b+c\right)+\left(b^2-ca\right)\left(a+c\right)+\left(c^2-ab\right)\left(b+a\right)}{\left(a+b\right)\left(a+c\right)\left(b+c\right)}\)

\(=\frac{a^2b+a^2c-b^2c-bc^2+ab^2+b^2c-a^2c-ac^2+ac^2-a^2b+bc^2-ab^2}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}\)

\(=\frac{0}{\left(a+b\right)\left(b+c\right)\left(a+c\right)}=0\)

Xét tứ giác MNPQ có :

\(\widehat{M}+\widehat{N}+\widehat{P}+\widehat{Q}=360^o\)

\(35^o+67^o+127^o+\widehat{Q}=360^o\)

\(229^o+\widehat{Q}=360^o\)

                 \(\widehat{Q}=360^o-229^o\)

                 \(\widehat{Q}=131^o\)

Vậy \(\widehat{Q}=131^o\)

Sửa đề : a² ( b - c ) + b² ( c - a ) + c² ( a - b ) = 0

Ta có

  \(a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)=0\)

\(\Leftrightarrow a^2b-a^2c+b^2c-b^2a-c^2b+c^2a=0\)

\(\Leftrightarrow\left(a^2b-b^2a\right)-\left(a^2c-b^2c\right)+\left(c^2a-c^2b\right)=0\)

\(\Leftrightarrow ab\left(a-b\right)-c\left(a-b\right)\left(a+b\right)+c^2\left(a-b\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(ab-ac-bc+c^2\right)=0\)

\(\Leftrightarrow\left(a-b\right)\left(b-c\right)\left(a-c\right)=0\)

\(\Leftrightarrow\begin{cases}a-b=0\\b-c=0\\a-c=0\end{cases}\Leftrightarrow\begin{cases}a=b\\b=c\\a=c\end{cases}\)

  Vậy a = b hoặc b = c hoặc a=c

\(3^4.5^4-\left(15^2+1\right)\left(15^2-1\right)\)

\(=\left(3.5\right)^4-\left[\left(15^2\right)^2-1\right]\)

\(=15^4-\left(15^4-1\right)=15^4-15^4+1=1\)

a) Ta có công thức hợp chất  \(XO_2\)

 \(XO_2=H_2.32=2.32=64\) 

Vậy phân tử khối của hợp chất \(XO_2\)là 64.

b) \(XO_2=64\)

Hay X + (16.2) = 64 => X = 32

Vậy nguyên tử khối của X là 32, X là nguyên tố lưu huỳnh và có kí hiệu hóa học là S.

?????

Áp dụng BĐT Cauchy-Schwarz ta có:

\(\left(a+b+c+1\right)^2=\left(a.1+\frac{1}{\sqrt{2}}.\sqrt{2}\left(b+c\right)+\frac{1}{\sqrt{2}}.\sqrt{2}\right)^2\)\(\le\left(a^2+1\right)\text{[}3+2\left(b+c\right)^2\text{]}\)

Khi đó cần CM BĐT : \(\frac{5}{16}\text{[}3+2\left(b+c\right)^2\text{]}\le\left(b^2+1\right)\left(c^2+1\right)\)

Hay: \(16b^2c^2+6\left(b^2+c^2\right)+1\ge20ab\)

BĐT trên đúng theo BĐT AM-GM: \(16b^2c^2+1\ge8bc,6\left(b^2+c^2\right)\ge12bc\)

Dấu '=' xảy ra khi và chỉ khi a=b=c=1/2

\(A=n^3+\left(n^3+3n^2+3n+1\right)+\left(n^3+6n^2+12n+8\right)\)

\(=3n^3+9n^2+15n+9=3\left(n^3+3n^2+5n+3\right)\)

Đặt \(B=n^3+3n^2+5n+1=n^3+n^2+2n^2+2n+3n+3\)

\(=n^2\left(n+1\right)+2n\left(n+1\right)+3\left(n+1\right)=n\left(n+1\right)\left(n+2\right)+3\left(n+1\right)\)

Ta thấy \(n\left(n+1\right)\left(n+2\right)⋮3\)( vì là tích của 3 số tự nhiên liên tiếp)

\(3\left(n+1\right)⋮3\Rightarrow B⋮3\Rightarrow A=3B⋮9\)