text: Recent questions tagged inequality
  
  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector.
  1 answer 46 views
  Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector.Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector. ...
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  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Find $x$ and $y$ if $(x, 3)=(2, x+y)$.
  1 answer 28 views
  Find $x$ and $y$ if $(x, 3)=(2, x+y)$.Find $x$ and $y$ if $(x, 3)=(2, x+y)$. ...
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  Prove that $A(B C)=(A B) C$.
  0 answers 28 views
  Prove that $A(B C)=(A B) C$. \text { If } A=\left\begin{array}{ll} 2 &amp; 6 \\ 7 &amp; 2 \end{array}\right B=\left\begin{array}{cc} -3 &amp; 5 \\ 0 &amp; 8 \e ...
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  Solve the system, if a is some real number:
  1 answer 37 views
  Solve the system, if a is some real number:Solve the system, if a is some real number: \begin{aligned} &amp;a x+y=1 \\ &amp;x+a y=1 \end{aligned} ...
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  If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$
  1 answer 50 views
  If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$ ...
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  $5 x+2(x+7)=14 x-7 .$ Find $x$
  1 answer 35 views
  $5 x+2(x+7)=14 x-7 .$ Find $x$$5 x+2(x+7)=14 x-7 . Find x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the value of x? 0 answers 45 views What is the value of x? What is the value of x? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Solve for x in \frac{5}{x}+\frac{1}{x}=\frac{2}{x}+2 1 answer 51 views Solve for x in \frac{5}{x}+\frac{1}{x}=\frac{2}{x}+2Solve for x in \dfrac{5}{x}+\dfrac{1}{x}=\dfrac{2}{x}+2 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68 1 answer 52 views Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Solve for x : \quad 2^{x}=\frac{1}{16} 1 answer 44 views Solve for x : \quad 2^{x}=\frac{1}{16}Solve for x : \quad 2^{x}=\dfrac{1}{16} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1 answer 61 views If x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} isIf x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1) 1 / 4 2) 41 3) 1 4) 17 / 7 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1 answer 75 views If x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} isIf x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1) 1 / 4 2) 41 3) 1 4) 17 / 7 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a is 1 answer 69 views If a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a isIf a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a is 1) less than 1 2) equal to 1 3) greater than 1 4) any real ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let a, b, c be positive real numbers such that a b c=1. Prove that 0 answers 85 views Let a, b, c be positive real numbers such that a b c=1. Prove thatLet a, b, c be positive real numbers such that a b c=1. Prove that$$ \frac{b+c}{\sqrt{a}}+\frac{a+c}{\sqrt{b}}+\frac{b+a}{\sqrt{c}} \geq \sqrt{a} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$. 0 answers 57 views Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$.Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}<1$$ 0 answers 58 views Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}<1$$Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}&lt;1$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$ 1 answer 53 views Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$ 0 answers 53 views Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$ 0 answers 63 views Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$ 0 answers 57 views If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$ 1 answer 57 views Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$. 1 answer 74 views Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$.Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$ 1 answer 89 views For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a>b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$ 1 answer 174 views Let$a>b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$Let$a&gt;b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If$x, y, z$are positive real numbers satisfying$x y z=32$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$ 0 answers 77 views If$x, y, z$are positive real numbers satisfying$x y z=32$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$If$x, y, z$are positive real numbers satisfying$x y z=32\$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$ ...
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explanation: The document contains multiple math questions and answers. Each question is a problem involving algebraic expressions, equations, or inequalities, and the answers provide solutions to these problems.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
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  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: Recent questions tagged inequality
  
  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector.
  1 answer 46 views
  Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector.Find $x$ and $y$ if $u=(x+y, x-3)$ is the zero vector. ...
  close
  
  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Find $x$ and $y$ if $(x, 3)=(2, x+y)$.
  1 answer 28 views
  Find $x$ and $y$ if $(x, 3)=(2, x+y)$.Find $x$ and $y$ if $(x, 3)=(2, x+y)$. ...
  close
  
  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Prove that $A(B C)=(A B) C$.
  0 answers 28 views
  Prove that $A(B C)=(A B) C$. \text { If } A=\left\begin{array}{ll} 2 &amp; 6 \\ 7 &amp; 2 \end{array}\right B=\left\begin{array}{cc} -3 &amp; 5 \\ 0 &amp; 8 \e ...
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  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  Solve the system, if a is some real number:
  1 answer 37 views
  Solve the system, if a is some real number:Solve the system, if a is some real number: \begin{aligned} &amp;a x+y=1 \\ &amp;x+a y=1 \end{aligned} ...
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  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$
  1 answer 50 views
  If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$If $a$ and $b$ are real numbers, solve the following equation: $(a+2) x-b=-2+(a+b) x$ ...
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  Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
  $5 x+2(x+7)=14 x-7 .$ Find $x$
  1 answer 35 views
  $5 x+2(x+7)=14 x-7 .$ Find $x$$5 x+2(x+7)=14 x-7 . Find x ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the value of x? 0 answers 45 views What is the value of x? What is the value of x? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Solve for x in \frac{5}{x}+\frac{1}{x}=\frac{2}{x}+2 1 answer 51 views Solve for x in \frac{5}{x}+\frac{1}{x}=\frac{2}{x}+2Solve for x in \dfrac{5}{x}+\dfrac{1}{x}=\dfrac{2}{x}+2 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68 1 answer 52 views Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68Show that 5.2^{x}+3.2^{x+2}=17.2^{x}, hence solve 5.2^{x}+3.2^{x+2}=68 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Solve for x : \quad 2^{x}=\frac{1}{16} 1 answer 44 views Solve for x : \quad 2^{x}=\frac{1}{16}Solve for x : \quad 2^{x}=\dfrac{1}{16} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1 answer 61 views If x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} isIf x is real, the maximum value of \dfrac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1) 1 / 4 2) 41 3) 1 4) 17 / 7 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1 answer 75 views If x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} isIf x is real, the maximum value of \frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7} is 1) 1 / 4 2) 41 3) 1 4) 17 / 7 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a is 1 answer 69 views If a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a isIf a, b, c are distinct +ve real numbers and a^{2}+b^{2}+c^{2}=1 then a b+b c+c a is 1) less than 1 2) equal to 1 3) greater than 1 4) any real ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let a, b, c be positive real numbers such that a b c=1. Prove that 0 answers 85 views Let a, b, c be positive real numbers such that a b c=1. Prove thatLet a, b, c be positive real numbers such that a b c=1. Prove that$$ \frac{b+c}{\sqrt{a}}+\frac{a+c}{\sqrt{b}}+\frac{b+a}{\sqrt{c}} \geq \sqrt{a} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$. 0 answers 57 views Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$.Given 3 positive reals$a, b$and$c$such that$a+b+c=1$, show that$a^{a} b^{b} c^{c}+a^{b} b^{c} c^{a}+a^{c} b^{a} c^{b} \leq 1$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}<1$$ 0 answers 58 views Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}<1$$Let$x$and$y$be positive real numbers satisfying$x^{3}+y^{3}=x-y$. Show that $$x^{2}+4 y^{2}&lt;1$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$ 1 answer 53 views Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$Given 3 positive reals$a, b$and$c_{,}$show that $$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$ 0 answers 53 views Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$Let$a$and$b$be positive real numbers. Show that $$\frac{1}{a}+\frac{1}{b} \geq \frac{4}{a+b}$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$ 0 answers 63 views Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$Let$a, b$and$c$be positive real numbers. Show that $$(a+b+c)(a b+b c+c a) \geq 9 a b c .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$ 0 answers 57 views If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$If$x$and$y$are positive real numbers, what is the minimum value of $$(2 x+3 y)\left(\frac{8}{x}+\frac{3}{y}\right) ?$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$ 1 answer 57 views Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$Find all positive real solutions$(a, b, c, d)$to the following system of equations: $$a+b+c+d=12, a b c d=27+a b+a c+a d+b c+b d+c d .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$. 1 answer 74 views Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$.Find all pairs of positive real numbers such that$a^{3}+b^{3}+1=3 a b$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$ 1 answer 89 views For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$For all positive real values$a, b, c$, prove that $$\frac{a^{2}}{b}+\frac{b^{2}}{c}+\frac{c^{2}}{d}+\frac{d^{2}}{a} \geq a+b+c+d .$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let$a>b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$ 1 answer 174 views Let$a>b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$Let$a&gt;b$be positive real numbers. What is the minimum value of $$a+\frac{1}{b(a-b)} ?$$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If$x, y, z$are positive real numbers satisfying$x y z=32$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$ 0 answers 77 views If$x, y, z$are positive real numbers satisfying$x y z=32$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$If$x, y, z$are positive real numbers satisfying$x y z=32\$, find the minimum value of $$x^{2}+4 x y+4 y^{2}+2 z^{2} .$$ ...
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains multiple math questions and answers. Each question is a problem involving algebraic expressions, equations, or inequalities, and the answers provide solutions to these problems.
  
  Label: multiple-question-and-answer