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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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