Calculate \(p=\dfrac{3x-2y}{3x+2y}\) know x , y satisfy x < 2y and \(9x^2+4y^2=20xy\)
Let x , y , z > 0 .Validity : \(p=\dfrac{x^{2018}+1}{x^{2018}+y^{2018}+z^{2018}+3}\) know \(x^3+y^3+z^3=3xyz\)
Let a, b, c be other numbers 0 satisfying : \(\left\{{}\begin{matrix}\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{a+b+c}\\a^3+b^3+c^3=2^9\end{matrix}\right.\)
Calculate \(p=a^{2019}+b^{2019}+c^{2019}\)
Let a , b , c be positive numbers . Prove that : \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ca}{c+a}\le\dfrac{1}{2}\left(a+b+c\right)\)
Let x , y , z be positive real numbers . Prove that \(\dfrac{x^2-z^2}{y+z}+\dfrac{y^2-x^2}{z+x}+\dfrac{z^2-y^2}{x+y}\ge0\)
Let a , b, c be a positive real numbers , prove that :
\(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge\dfrac{1}{\sqrt{2}}\left(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}\right)\)
What is the condition for the equality ?
thank you
Solve the following system of equations :
\(\left\{{}\begin{matrix}4\sqrt{3x+4y}+\sqrt{8-x+y}=23\\3\sqrt{8-x+y}-2\sqrt{38+6x-13y}=5\end{matrix}\right.\)
Given the square triangle ABC at A , the AD curve ( D of BC ) .
Prove that \(\dfrac{\sqrt{2}}{AD}=\dfrac{1}{AB}+\dfrac{1}{AC}\)
A trapezoid has an area of 1 . Ask the diagonal of this trapezoid is the smallest .
For the acute triangle ABC , the circle (O) has a fixed BC ( BC ≠ 2R ) , H is the center. Determine the position of A to: S = HA + HB + HC is the maximum value.
Given the following equation :
\(2x^4-4x^3+\left(4-a\right)x^2+\left(a-2\right)x+a-a^2=0\)
Prove that the equation has only one negative root when \(a>1\)
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Solve the following equation with the parameter m :
\(\dfrac{x+1}{x-m}+\dfrac{x+m}{x-1}=2\)
solve the equation :
\(\dfrac{1}{x\left(x+2\right)}+\dfrac{1}{\left(x+2\right)\left(x+4\right)}+\dfrac{1}{\left(x+4\right)\left(x+6\right)}=\dfrac{1}{9}\)
solving the equation with a , b , c is the parameter : ( abc is different 0 )
\(\dfrac{x-b-c}{a}+\dfrac{x-a-c}{b}+\dfrac{x-a-b}{c}=3\)