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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.

Level 4

For positive real numbers $$a,b,c,d,$$ find the minimum value of the following expression:

$\displaystyle (a+b+c+d)\Bigl( \dfrac{25}{a} + \dfrac{36}{b} +\dfrac{81}{c} +\dfrac{144}{d} \Bigr).$

Let $$a,b,c,$$ and $$d$$ be real numbers. Find the maximum possible value of $\min\{a-b^2,\ b-c^2,\ c-d^2,\ d-a^2\}.$

Consider all positive reals $$a$$ and $$b$$ such that

$ab (a+b) = 2000.$

What is the minimum value of

$\frac{1}{a} + \frac{1}{b} + \frac{1}{a+b} ?$

$$a, b$$ and $$c$$ are positive real numbers satisfying $$a + b + c = 100$$. The maximum value of

$a + \sqrt{ab} + \sqrt[3]{abc}$

can be expressed as $$\dfrac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers. What is the value of $$m+n$$?

$$x_1, x_2, x_3, x_4$$ and $$x_5$$ are reals such that $$x_1+x_2+x_3+x_4+x_5=8$$ and $$x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=16.$$ What is the largest possible value of $$x_5?$$

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