Tomi's Challenges - #6

As we'll see, this problem requires an understanding of how graphs work.

Let $2021a+\sin^4(\frac{1}{2021^2a^6}x)-|x^2-2021^{2021a^{12}}|=0$ . We have:

$2021a+\sin^4(\frac{1}{2021^2a^6}x)=|x^2-2021^{2021a^{12}}|$

There is no point in trying to manipulate this equation any further, so let's think about how this equation looks visually. It's an intersection of a trigonometric function (which looks like similar to sine, but it's restricted to positive values) and an "absolute parabola". This absolute parabola is $x^2$ , but translated 1 unit down $2021^{2021a^{12}}$ times, and everything below the x-axis is mirrored to be above the x-axis. We need to find how many times these two graphs intersect. Notice that since $2021^{2021a^{12}}$ (which lifts the remainder of the parabola up) is much bigger than $2021a$ (which lifts the trigonometric function up), these 2 graphs will always intersect at least 4 times, which should be clear intuitively. Another important characteristic about this trigonometric function is that regardless of $a$ , its frequency is so small that there is no chance that the parabola will ever intersect one of its waves more than $4$ times, just because of how fast it grows - i.e. how high its derivative is, compared to the trigonometric function. So from this we conclude that the number of intersections is $4$ .

Here is how it should look graphically. Note that I put much smaller values than the ones in this exercise, for presentation purposes.