×
Back to all chapters

# 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.

# 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}

×