Geometric Inequalities

Geometric Inequalities: Level 4 Challenges


a2+1+b2+16+c2+49\large \sqrt{a^2+1}+\sqrt{b^2+16}+\sqrt{c^2+49}

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

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, AA and BB 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 CC. Find the maximum value of the expression A+BC\dfrac{A+B}{C}, the answer is of the form X\sqrt{X} and so find XX.

Given constants λ,μ+\lambda ,\mu \in { \Re }^{ + }, consider the set of points x,yx, y such that 2x+y+λ+xy+μ4. 2 \leq \left| x+y+\lambda \right| + \left| x-y+\mu \right| \leq 4.

What is the area of this region ?

x43x26x+13x4x2+1\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 a\sqrt{a} , find the value of a2a^2.


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