Algebra

Classical Inequalities

Classical Inequalities: Level 4 Challenges

         

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

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

Let a,b,c,a,b,c, and dd be real numbers. Find the maximum possible value of min{ab2, bc2, cd2, da2}.\min\{a-b^2,\ b-c^2,\ c-d^2,\ d-a^2\}.

Consider all positive reals aa and bb such that

ab(a+b)=2000. ab (a+b) = 2000.

What is the minimum value of

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

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

a+ab+abc3 a + \sqrt{ab} + \sqrt[3]{abc}

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

x1,x2,x3,x4x_1, x_2, x_3, x_4 and x5x_5 are reals such that x1+x2+x3+x4+x5=8x_1+x_2+x_3+x_4+x_5=8 and x12+x22+x32+x42+x52=16.x_1^2+x_2^2+x_3^2+x_4^2+x_5^2=16. What is the largest possible value of x5?x_5?

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