Floor Function Problem...

Solve the following equation:

$\displaystyle \left\lfloor x+\frac{7}{3} \right\rfloor^2-\left\lfloor x-\frac{9}{4} \right\rfloor=16$

Note: $\displaystyle \lfloor x \rfloor$ denotes the largest integer not greater than $x$. This function, referred to as the floor function, is also called the greatest integer function, and its value at $x$ is called the integral part or integer part of $x$.

Analytic Geometry: Orthogonal Trajectories

The problem is as follows:

a) Find the family of circles centered on the $y$-axis, that pass through the points $(\pm a,0)$, where [math]0<a\le r\in\mathbb{R}[/math].

b) Find the family of curves orthogonal to the family of circles found in part a).

Hint 1: Express the family of circles from part a) in the form:

[math]F(x,y)=C[/math]

Hint 2: It can be shown that the orthogonal trajectories must satisfy:

[math]\frac{\partial F}{\partial y}dx-\frac{\partial F}{\partial x}dy=0[/math]

16x²y²-48x²y+24xy²+100x²+16y²-72xy+150x-48y+100=28

Determine the pair(s) of real numbers $(a,\,b)$ that satisfy the equation $16x^2y^2-48x^2y+24xy^2+100x^2+16y^2-72xy+150x-48y+100=28$.

At first glance,  this seems like we need to use the modular arithmetic method to solve for $(x,\,y)$ but wait a minute! We need to find not the integers but real for $(x,\,y)$, so nope, modular arithmetic is out of the question...

Fun With "Normal Lines"

The curve in the figures above is the parabola $y=x^2$. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. Five normal lines are shown in the figures above.

For a while, the $x$-coordinate of the second quadrant intersection of a normal line with the parabola gets smaller as the $x$-coordinate of the first quadrant intersection gets smaller. But eventually a normal line's second quadrant intersection gets as small as it can get.

The extreme normal line is shown as a thick red line in the figures above. Once the normal lines pass the extreme normal line, the $x$-coordinate of the second quadrant intersections with the parabola start to increase.

The figures above show two pairs of normal lines. The two normal lines of a pair have the same second quadrant intersection with the parabola, but one is above the extreme normal line (in the first quadrant) and the other is below it.

(a) Find the normal line pair associated with a particular second quadrant intersection with the parabola.

(b) Hence, or otherwise, find the equation of the extreme normal line.

(c) Find the normal line that traps the smallest area between it and the parabola.