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\(\text{Σ}\frac{a}{b+2c+3d}=\text{Σ}\frac{a^2}{ab+2ac+3ad}\ge\frac{\left(a+b+c+d\right)^2}{6\left(ab+bc+cd+ad\right)}\)
\(=\frac{\left(a+b\right)^2+\left(c+d\right)^2+2\left(a+b\right)\left(c+d\right)}{6\left(ab+bc+cd+ad\right)}=\frac{a^2+c^2+b^2+d^2+2ab+2cd+2\left(a+b\right)\left(c+d\right)}{6\left(ab+bc+cd+ad\right)}\)
\(\ge\frac{4\left(ab+bc+cd+ad\right)}{6\left(ab+bc+cd+ad\right)}=\frac{2}{3}\)
Dấu = xảy ra khi a=b=c=d
Áp dụng BĐT Cauchy-Schwarz ta có:
\(\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c}\)
\(=\frac{a^2}{ab+2ac+3ad}+\frac{b^2}{bc+2bd+3ab}+\frac{c^2}{cd+2ac+3bc}+\frac{d^2}{ad+2bd+3cd}\)
\(\ge\frac{\left(a+b+c+d\right)^2}{4.\left(ab+ad+bc+bd+ca+cd\right)}\)\(\ge\frac{\left(a+b+c+d\right)^2}{\frac{3}{2}.\left(a+b+c+d\right)^2}=\frac{2}{3}\)
Dấu "=" xảy ra \(\Leftrightarrow a=b=c=d\)
\(P=4a+3b+\frac{c^3}{\left(a-b\right)b}\)
\(=\left[\left(a-b\right)+b+\frac{c^3}{\left(a-b\right)b}\right]+3b+3a\)
\(\ge3c+3b+3a=3\left(a+b+c\right)=12\)
Dấu "=" xảy ra tại \(a=2;b=1;c=1\)
\(a+\frac{3}{5}=\frac{b-2}{3}=c-\frac{1}{7}\)
\(a+\frac{3}{5}=b-\frac{2}{3},b-\frac{2}{3}=c-\frac{1}{7}\)
\(b=\frac{15r+19}{15}\)
\(c=\frac{35r+26}{35}\)\(;r\in R\)
\(\frac{a+3}{5}=\frac{b-2}{3}=\frac{c-1}{7}=\frac{3a+9}{15}=\frac{5b-10}{15}=\frac{7c-7}{49}\)
Áp dụng tính chất dãy tỉ số bằng nhau:
\(\frac{a+3}{5}=\frac{b-2}{3}=\frac{c-1}{7}=\frac{3a+9}{15}=\frac{5b-10}{15}=\frac{7c-7}{49}\)
\(=\frac{3a+9-5b+10+7c-7}{15-15+49}=\frac{86+12}{49}=2\)
=>a=2.5-3=7;b=2.3+2=8;c=2.7+1=15
B1:
A.1+1=2
B.6+2+5=13
C.6+4+6+2+7=25
D.6+4+7+8=25
B2:
A.x+2=3
x=3-2
x=1
B.x+5=4+6
x+5=10
x=10-5
x=5
C.7=4+x
x=7-4
x=3
D.4+x={3+5+7}+{3+5}
4+x=15+8
4+x=23
x=23-4
x=19
Bài 1 :
A.1+1=2
B.6+2+5=13
C.6+4+6+2+7=25
D.6+4+7+8=25
Bài 2 :
A.x+2=3
x=3-2
x=1
B.x+5=4+6
x+5=10
x=10-5
x=5
C.7=4+x
x=7-4
x=3
D.4+x={3+5+7}+{3+5}
4+x=15+8
4+x=23
x=23-4
x=19
Chúc bạn học tốt nha ~_~
\(\frac{a}{4}=\frac{b}{6}\)\(,\)\(\frac{b}{5}=\frac{c}{8}\)
\(\Rightarrow\left(a=\frac{2b}{3}\right)\)\(,\)\(\left(b=\frac{5c}{8}\right)\)
\(\Rightarrow3a=2b\)\(,\)\(8b=5c\)
\(\Rightarrow b=\frac{3a}{2}\)\(,\)\(c=\frac{12a}{5}\)
\(\Rightarrow a=10\)\(,\)\(b=15\)\(,\)\(c=24\)
\(\frac{a}{4}=\frac{b}{6}.\frac{b}{5}=\frac{c}{8}\)
\(\Rightarrow\left(3a=2b,8b=5c\right)\)
\(\Rightarrow b=\frac{3a}{2}.c=\frac{12a}{5}\)
\(\Rightarrow a=10,b=15,c=24\)