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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.

Level 4

         

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

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

Find the number of isosceles triangles with integer sides when no side exceeds 1994. Include equilateral triangles in your count.

In a right angled triangle, \(A\) and \(B\) denote the lenghts of the medians that belong to the legs of the triangle, and the length of the median belonging to the hypotenuse is \(C\). Find the maximum value of the expression \(\dfrac{A+B}{C}\), the answer is of the form \(\sqrt{X}\) and so find \(X\).

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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