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Lời giải:
Gọi biểu thức đã cho là $A$
Đặt $2a-5b=x; 3b-7c=y; c-6a=z$
$\Rightarrow x+y+z=-2(2a+b+3c)$ chẵn
$A=|x|+|y|+|z|$
$A^2=(|x|+|y|+|z|)^2=x^2+y^2+z^2+2|xy|+2|yz|+2|xz|$
$=(x+y+z)^2-2xy-2yz-2xz+2|xy|+2|yz|+2|xz|$
chẵn do $x+y+z$ chẵn
$A^2$ chẵn kéo theo $A$ chẵn (đpcm)
Lời giải:
Đặt $a-b=x; b-c=y; c-d=z; d-a=t$ thì $x+y+z+t=0$
$\Rightarrow x+y=-(z+t)$
$\Rightarrow (x+y)^2=(z+t)^2$
Đặt $A=|a-b|+|b-c|+|c-d|+|d-a|=|x|+|y|+|z|+|t|$
$A^2=(|x|+|y|+|z|+|t|)^2$
$=(|x|+|y|)^2+(|z|+|t|)^2+2(|x|+|y|)(|z|+|t|)$
$=x^2+y^2+z^2+t^2+2|xy|+2|zt|+2(|x|+|y|)(|z|+|t|)$
$=(x+y)^2+(z+t)^2-2xy-2zt+2|xy|+2|zt|+2(|x|+|y|)(|z|+|t|)$
$=2(z+t)^2-2xy-2zt+2|xy|+2|zt|+2(|x|+|y|)(|z|+|t|)$ chẵn
$A^2$ chẵn kéo theo $A$ chẵn (đpcm)
Đã giải tại đây:
https://hoc24.vn/cau-hoi/cho-cac-so-nguyen-abcd-chung-minh-rang-tong-a-bb-cc-dd-a-luon-la-so-chantks-mn.1408463365507
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Ta có : 2a = 3b => \(\frac{a}{3}=\frac{b}{2}\)
5b = 7c => \(\frac{b}{7}=\frac{c}{5}\)
=> \(\frac{a}{3}=\frac{b}{2};\frac{b}{7}=\frac{c}{5}\)
+) \(\frac{a}{3}=\frac{b}{2}\Rightarrow\frac{a}{21}=\frac{b}{14}\)
+) \(\frac{b}{7}=\frac{c}{5}\Rightarrow\frac{b}{14}=\frac{c}{10}\)
=> \(\frac{a}{21}=\frac{b}{14}=\frac{c}{10}\)
=> \(\frac{3a}{63}=\frac{7b}{98}=\frac{5c}{50}\)
Áp dụng t/c dãy tỉ số bằng nhau ta có : \(\frac{3a}{63}=\frac{7b}{98}=\frac{5c}{50}=\frac{3a+5c-7b}{63+50-98}=\frac{30}{15}=2\)
Từ đó suy ra a = 2.21 = 42,b = 2.14 = 28,c = 2.10 = 20
Ta có:\(2a=3b\)\(\Rightarrow\frac{a}{3}=\frac{b}{2}\)\(\Rightarrow\frac{a}{21}=\frac{b}{14}\)
\(5b=7c\)\(\Rightarrow\frac{b}{7}=\frac{c}{5}\)\(\Rightarrow\frac{b}{14}=\frac{c}{10}\)
Suy ra:\(\frac{a}{21}=\frac{b}{14}=\frac{c}{10}\)
Đặt\(\frac{a}{21}=\frac{b}{14}=\frac{c}{10}=k\)
\(\Rightarrow\hept{\begin{cases}a=21k\\b=14k\\c=10k\end{cases}}\)
Mà\(3a+5c-7b=30\)
\(\Rightarrow3.21k+5.10k-7.14k=30\)
\(\Leftrightarrow63k+50k-98k=30\)
\(\Leftrightarrow15k=30\)
\(\Leftrightarrow k=2\)
\(\Rightarrow\hept{\begin{cases}a=2.21=42\\b=2.14=28\\c=2.10=20\end{cases}}\)
Vậy\(\hept{\begin{cases}a=42\\b=28\\c=20\end{cases}}\)
Linz
2a-8b+3b-7c+c-6a=-4a-2b-6c=2(-2a-b-3c) luôn là số chẵn