Algebra

Classical Inequalities

Applying the Arithmetic Mean Geometric Mean Inequality

         

What is the smallest possible value of the expression x2+361x\frac{x^2+361}{x} for positive real xx?

What is the minimum value of y=x2x9y=\frac{x^2}{x-9} when x>9x>9?

How many ordered pairs of real numbers satisfy

36x2+y+36y2+x=26? 36 ^ { x^2 + y} + 36 ^{ y^2 + x } = \frac{2} {\sqrt{6} } ?

Let y1,y2,y3,,y8 y_1, y_2, y_3, \ldots, y_{8} be a permutation of the numbers 1,2,3,,8 1, 2, 3, \ldots, 8 . What is the minimum value of i=18(yi+i)2 \displaystyle \sum_{i=1}^8 (y_i + i )^2 ?

If aa and bb are positive numbers such that a1,b1a\ne 1, b\ne 1 and a+b1,a+b \neq 1, what is the order relation of the following 33 expressions A,BA, B and C?C? A=1loga2+1logb2,B=2(1loga+b21),C=2(1+1logab21loga+b2)\begin{aligned} A &= \frac{1}{\log_a 2}+\frac{1}{\log_b 2}, \\ B &= 2\left(\frac{1}{\log_{a+b} 2}-1\right), \\ C &= 2\left(1+\frac{1}{\log_{ab} 2}-\frac{1}{\log_{a+b} 2}\right) \end{aligned}

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