Algebra

# Geometric Inequalities: Level 4 Challenges

$\large \sqrt{a^2+1}+\sqrt{b^2+16}+\sqrt{c^2+49}$

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