\(\Leftrightarrow\frac{1}{1+a}+\frac{a}{1+a}+\frac{2b}{21+2b}+\frac{21}{21+2b}\le\frac{4c}{4c+27}+\frac{a}{1+a}+\frac{2b}{21+2b}\)

\(\Leftrightarrow2\le\frac{1}{1+\frac{1}{a}}+\frac{1}{1+\frac{21}{2b}}+\frac{1}{1+\frac{27}{4c}}\)

Đặt \(\left(\frac{1}{a};\frac{21}{2b};\frac{27}{4c}\right)=\left(x;y;z\right)\)

\(\Leftrightarrow\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge2\)

\(\Leftrightarrow\frac{1}{1+x}\ge1-\frac{1}{1+y}+1-\frac{1}{1+z}=\frac{y}{1+y}+\frac{z}{1+z}\ge2\sqrt{\frac{yz}{\left(1+y\right)\left(1+z\right)}}\)

Tương tự: \(\frac{1}{1+y}\ge2\sqrt{\frac{zx}{\left(1+z\right)\left(1+x\right)}}\) ; \(\frac{1}{1+z}\ge2\sqrt{\frac{xy}{\left(1+x\right)\left(1+y\right)}}\)

Nhân vế với vế: \(1\ge8xyz\Rightarrow xyz\le\frac{1}{8}\)

\(\Leftrightarrow\frac{1}{a}.\frac{21}{2b}.\frac{27}{4c}\le\frac{1}{8}\Leftrightarrow abc\ge567\)

Dấu "=" xảy ra khi \(\frac{1}{a}=\frac{21}{2b}=\frac{27}{4c}=\frac{1}{2}\Rightarrow\left(a;b;c\right)=\left(2;21;\frac{27}{2}\right)\)

a) Ta được: 18 . 21 = 27 . 14

b) Từ 4 số: 14; 18; 21; 27, ta có đẳng thức sau: 18 . 27 = 21 . 14, ta lập được các tỉ lệ thức:

\(\frac{{18}}{{27}} = \frac{{14}}{{21}};\frac{{18}}{{14}} = \frac{{27}}{{21}};\frac{{14}}{{18}} = \frac{{21}}{{27}};\frac{{21}}{{14}} = \frac{{27}}{{18}}\)

a: 18*21=27*14

b: 18/27=14/21; 18/14=27/21; 27/18=21/14; 14/18=21/27

Đáp án đúng là: A

Dãy số 21; – 3; – 27; – 51; – 75 lập thành một cấp số cộng có số hạng đầu là u₁ = 21 và công sai d = – 24.

Ta có: \(\frac{54}{11}.\frac{121}{27}< n< \frac{100}{21}:\frac{25}{126}\)

\(\Rightarrow\frac{2.11}{1.1}< n< \frac{100}{21}.\frac{126}{25}\)

\(\Rightarrow22< n< 24\)

\(\Rightarrow n=23\)

Ta có:

\(\frac{54}{11}\cdot\frac{121}{27}=\frac{54\cdot121}{11\cdot27}=22\)

\(\frac{100}{21}:\frac{25}{126}=\frac{100}{21}\cdot\frac{126}{25}=\frac{100\cdot126}{21\cdot25}=24\)

\(\Rightarrow22< n< 24\)

\(\Rightarrow x=23\)

So sánh với 1:

\(\frac{19}{15}>1\)

\(\frac{21}{27}< 1\)

Vậy: \(\frac{19}{15}< \frac{21}{27}\)

Ta có : \(\frac{19}{15}+1=\frac{34}{15}\)

\(\frac{21}{27}+1=\frac{46}{27}\)

Vì \(\frac{34}{15}>\frac{46}{27}\Rightarrow\frac{19}{15}>\frac{21}{27}\)

Ta có :

\(\frac{19}{15}\)là phân số có tử lớn hơn mẫu nên \(\frac{19}{15}>1\)

\(\frac{21}{27}\)la phân số có tử bé hơn mẫu nên \(\frac{21}{27}< 1\)

Vậy \(\frac{19}{15}>\frac{21}{27}\)

\(\frac{29-x}{21}+\frac{27-x}{23}+\frac{25-x}{25}+\frac{23-x}{27}+\frac{21-x}{29}=-5.\)

\(\left(\frac{29-x}{21}+1\right)+\left(\frac{27-x}{23}+1\right)+\left(\frac{25-x}{25}+1\right)+\left(\frac{23-x}{27}+1\right)+\left(\frac{21-x}{29}+1\right)\)\(=0\)

\(\frac{50-x}{21}+\frac{50-x}{23}+\frac{50-x}{25}+\frac{50-x}{27}+\frac{50-x}{29}=0\)

\(\left(50-x\right).\left(\frac{1}{21}+\frac{1}{23}+\frac{1}{25}+\frac{1}{27}+\frac{1}{29}\right)=0\)

=> 50 - x = 0 \(\left(\frac{1}{21}+\frac{1}{23}+\frac{1}{25}+\frac{1}{27}+\frac{1}{29}\ne0\right)\)

=> x = 50