Waste less time on Facebook — follow Brilliant.
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.
For positive real numbers \(a,b,c,d,\) find the minimum value of the following expression:
\[\displaystyle (a+b+c+d)\Bigl( \dfrac{25}{a} + \dfrac{36}{b} +\dfrac{81}{c} +\dfrac{144}{d} \Bigr). \]
by
Aditya Raut
Let \(a,b,c,\) and \(d\) be real numbers. Find the maximum possible value of \[\min\{a-b^2,\ b-c^2,\ c-d^2,\ d-a^2\}.\]
by
Sompong Chuisurichy
Consider all positive reals \(a\) and \(b\) such that
\[ ab (a+b) = 2000. \]
What is the minimum value of
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{a+b} ?\]
by
Calvin Lin
\(a, b\) 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 \(m+n\)?
by
Calvin Lin
\(x_1, x_2, x_3, x_4\) 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 \(x_5?\)
by
Victor Loh