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Classical Inequalities

Challenge Quizzes

Classical Inequalities: Level 4 Challenges


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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