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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

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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