Cho a,b,ca,b,c là các số với |a|,|b|,|c|≤1|a|,|b|,|c|≤1chứng minh rằng, nếu a,b,ca,b,c thỏa mãn: ...
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Cho a,b,c" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-size: 13.696px; vertical-align: baseline; display: inline; line-height: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(0, 0, 0); font-family: Arial, 'Liberation Sans', 'DejaVu Sans', sans-serif; position: relative; background: transparent;">a,b,c là các số với |a|,|b|,|c|≤1" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-size: 13.696px; vertical-align: baseline; display: inline; line-height: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; color: rgb(0, 0, 0); font-family: Arial, 'Liberation Sans', 'DejaVu Sans', sans-serif; position: relative; background: transparent;">|a|,|b|,|c|≤1
chứng minh rằng, nếu a,b,c" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-size: 13.696px; vertical-align: baseline; display: inline-table; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; background: transparent;">a,b,c thỏa mãn:
a2+b2+c2=1−2abc" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-size: 13.696px; vertical-align: baseline; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; background: transparent;">a2+b2+c2=1−2abc
thì
a+b+c=2(1−a)(1−b)(1−c)2+1" role="presentation" style="margin: 0px; padding: 0px; border: 0px; font-size: 13.696px; vertical-align: baseline; display: inline; line-height: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative; background: transparent;">a+b+c=2(1−a)(1−b)(1−c)2−−−−−−−−−−−−−−−−−√+1
\(\left(\frac{1}{x^2}-1\right)\left(\frac{1}{y^2}-1\right)\left(\frac{1}{z^2}-1\right)=\frac{\left(1-x^2\right)\left(1-y^2\right)\left(1-z^2\right)}{x^2y^2z^2}\)
\(=\frac{\left(1+x\right)\left(1-x\right)\left(1+y\right)\left(1-y\right)\left(1+z\right)\left(1-z\right)}{x^2y^2z^2}\)
\(\ge\frac{\left(x+y+z+x\right)\left(x+y+z-x\right)\left(x+y+z+y\right)\left(x+y+z-y\right)\left(x+y+z+z\right)\left(x+y+z-z\right)}{x^2y^2z^2}\)
\(\ge\frac{4\sqrt[4]{x^2yz}.2\sqrt{yz}.4\sqrt[4]{xy^2z}.2\sqrt{zx}.4\sqrt[4]{xyz^2}.2\sqrt{xy}}{x^2y^2z^2}\)
\(=\frac{512x^2y^2z^2}{x^2y^2z^2}=512\)