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Classical Inequalities

Your destination for questions of the form "do we know if this expression is always greater than this other expression?" Explore what humans know about mathematical inequalities.

Challenge Quizzes

Applying the Arithmetic Mean Geometric Mean Inequality

         

What is the smallest possible value of the expression \[\frac{x^2+361}{x}\] for positive real \(x\)?

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

How many ordered pairs of real numbers satisfy

\[ 36 ^ { x^2 + y} + 36 ^{ y^2 + x } = \frac{2} {\sqrt{6} } ? \]

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

If \(a\) and \(b\) are positive numbers such that \(a\ne 1, b\ne 1\) and \(a+b \neq 1,\) what is the order relation of the following \(3\) expressions \(A, B\) and \(C?\) \[\begin{align} A &= \frac{1}{\log_a 2}+\frac{1}{\log_b 2}, \\ B &= 2\left(\frac{1}{\log_{a+b} 2}-1\right), \\ C &= 2\left(1+\frac{1}{\log_{ab} 2}-\frac{1}{\log_{a+b} 2}\right) \end{align}\]

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