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# Geometric Inequalities

Given 5 sticks of length 1, 3, 5, 9, and 10, how many distinct triangles can be formed? Learn the techniques and develop an intuition for working with geometric inequalities.

\[\sqrt{a^2+1}+\sqrt{b^2+16}+\sqrt{c^2+49} \]

For \(a,b,c \in \mathbb R \) such that \(a+b+c=5\), what is the minimum value of the expression below?

Given constants \(\lambda ,\mu \in { \Re }^{ + }\), consider the set of points \(x, y \) such that \[ 2 \leq \left| x+y+\lambda \right| + \left| x-y+\mu \right| \leq 4.\]

What is the area of this region ?

\[\large \sqrt{x^4-3x^2-6x+13} - \sqrt{x^4-x^2+1}\]

If the maximum value of the function above can be expressed as \(\sqrt{a}\) , find the value of \(a^2\).

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