1/cho cac so duong a,b,c,x,y,z thoa man ax=by=cz=10
chung minh rang [a+b+c][x+y+z]>=90
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cho x,y,z la cac so nguyen duong va x+y+z la so le, cac so thuc a,b,c thoa man (a-b)/x=(b-c)/y=(a-c)/z. chung minh rang a=b=c
cho 3 so thuc x,y,z khac khong va thoa man hai dieu kien \(ax^3=by^3=cz^3\) va \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\)
chung minh rang : \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)
Ta có \(ax^3=by^3=cz^3\Leftrightarrow\dfrac{ax^2}{\dfrac{1}{x}}=\dfrac{by^2}{\dfrac{1}{y}}=\dfrac{cz^2}{\dfrac{1}{z}}=\dfrac{ax^2+by^2+cz^2}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}=ax^2+by^2+cz^2\Leftrightarrow\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{ax^3}=\sqrt[3]{by^3}=\sqrt[3]{cz^3}=\dfrac{\sqrt[3]{a}}{\dfrac{1}{x}}+\dfrac{\sqrt[3]{b}}{\dfrac{1}{y}}+\dfrac{\sqrt[3]{c}}{\dfrac{1}{z}}=\dfrac{\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}}{\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)Vậy \(\sqrt[3]{ax^2+by^2+cz^2}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}\)
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cho a,b,c,x,y,z la cac so nguyen duong thoa man a^x=bc;b^y=ac;c^z=ab. chung minh xyz-x-y-z=2
cho cac so thuc x,y,z,a,b,c thoa man \(\frac{x}{a}\)=\(\frac{y}{b}\)=\(\frac{z}{c}\)
CMR \(\frac{x^2+y^2+z^2}{\left(ax+by+cz\right)^2}\)= \(\frac{1}{a^2+b^2+c^2}\)
Ax+By=Cz voi cac dieu kien A,B,C,x,y,z deu la cac so nguyen duong, trong do x,y,z lon hon 2con A,B,C có cùng boi số chung nhỏ nhất
ai giải được mình cho 3 like
cho a,b,c,x,y,zkhac \
x/a=y/b=z/c chung minh rang (x^2+y^2+z^2)/(ax+by+cz)^2=1/(a^2+b^2+c^2)
cho a,b,c,x,y,z thỏa mãn: ax+by=c, by+cz=a, cz+ax=b, x,y,z khác -1, (a+b+c) khác 0. Tính P=1/(x+1)+1/(y+1)+1/(z+1)
Ta có ax + by = c ; by + cz = a
<=> cz - ax = a - c (1)
mà cz + ax = b (2)
Từ (1) và (2) => \(cz=\frac{a-c+b}{2}\Rightarrow z=\frac{a-c+b}{2c}\Rightarrow z+1=\frac{a+b+c}{2c}\)
=> \(\frac{1}{z+1}=\frac{2c}{a+b+c}\)
Tương tự ta có \(\frac{1}{x+1}=\frac{2a}{a+b+c}\); \(\frac{1}{y+1}=\frac{2b}{a+b+c}\)
=> P = \(\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}=\frac{2a}{a+b+c}+\frac{2b}{a+b+c}+\frac{2c}{a+b+c}=2\)
cho x, y , z la cac so nguyen thoa man x . y - x. z + y.z - z^2 +1 =0 chung minh rang x+ y =0
cho cac so x,y,z thoa man x/2013=y/2014=z/2015 chung minh rang 4(x-y)(y-z)=(z-x)^2