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Algebra

Classical 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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