@the-blade-dancer

trigonometry is a very i n t e r e s t i n g subject, but my math teacher said that it's the main thing that "murders" high school students' grades and self-esteem. It's a good idea to start getting the concepts when you're in middle school.

@the-blade-dancer I suppose you can prove this with coordinate bashing, that it is impossible for a line to touch a circle at more than 2 points.

@mathnerd_101 Glad that you figured it out!

@bulba_bulbasaur

@energizedpanda In the March 26 Ask Math Anything livestream, Prof. Loh actually explained why the volume of a sphere is \(\frac{1}{3}^{\text{rd}}\) its surface area!

It has to do with the fact that you can chop the volume into infinitesimally-small pyramids where the bases lie on the surface of the sphere and the opposite corners all lie at the center of the sphere. Then, since the volume of a pyramid is just \(\frac{1}{3} \text{ ( area of base ) } \times \text{ height },\) you just take the surface area, multiply by the height of each pyramid, which is the same for each pyramid, \(r,\) and multiply by \(\frac{1}{3}\) to get the volume of the sphere!

AMC 8 2013 Problem 25 I remember doing this.

@tidyboar

@The-Blade-Dancer The factor of \(2^{19}\) comes from all the even numbers in the factorial \(N!,\) like \(2, 4, 6, 8, 12, 14, \ldots\) etc., in addition to extra \(2\)'s that come from the powers of \(2\) like \(4, 8, 16.\)

Actually, though, you know that \(N\) must equal \(22,\) since we have two \(11\)'s (from the fact that \(11^2\) is a factor). One is from \(11,\) and the other is from \(22.\) We know that \(N\) can't be larger than \(22,\) so \(\boxed{N=22}.\)

And then \(N!\) (N factorial) would be \(22! = 1 \times \textcolor{red}{2} \times 3 \times \textcolor{red}{4} \times 5 \times \textcolor{red}{6} \times \textcolor{blue}{7} \times \textcolor{red}{8} \times 9 \times \textcolor{red}{10} \times \textcolor{yellow}{11} \times \textcolor{red}{12} \times 13 \times \textcolor{purple}{14} \times 15 \times \textcolor{red}{16} \times 17 \times \textcolor{red}{18} \times 19 \times \textcolor{red}{20} \times \textcolor{blue}{21} \times \textcolor{orange}{22}.\)

I tried to color-code so that all of the \(\textcolor{red}{\text{red}}\) numbers are even and contribute factors of \(2\) (this is where you get the \(2^{19}\)), the \(\textcolor{blue}{\text{blue}}\) numbers contribute factors of \(7\), and the yellow numbers contribute factors of \(11.\) Since \(\textcolor{purple}{14}\) is a multiple of both \(2\) and \(7,\) I made it purple. And since \(22\) is a multiple of both \(11\) and \(2,\) I made it orange.

@authenticwasp How many digits can the number have?

Yeah I also found a=3 and b=2. I'm pretty sure there's no more solutions but I don't know how to prove that

@RZ923 That's a very good question! I never learned the proof, but I believe it involves induction. You can more easily prove that Pick's Theorem works for simple shapes like triangles and rectangles. Using these smaller shapes, you create any polygon whose vertices lie on lattice points (points with integer coordinates). Sorry that I don't know the proof right now, but I'll try to look into it someday!

Try using a tablet, i guess?

lol you have a point there

Hi @RZ923!
Well, that's a nice question and is a very deep concept. You can read about it here.

@smartlion You can also bring these questions to the Ask Math Anything live stream and ask Prof. Loh! That is probably the best way, since we (the moderators) are often too busy with other work to answer non-course-related math questions.

@nastya Thanks

Hi @The-Darkin-Blade!

The method of converting repeating decimals to fractions is just one of the algebraic methods of working with rational fractions. It is just an algorithm that tells us what to do. It is the same as the method of solving equations, inequalities or changing fractions to a reduced form. Honestly, there is not only one algorithm of converting repeating decimals to fractions - there are lots of them. Some of them are harder and require some background, some of them are easier and do not require any prior knowledge.

So, once again, it's just an algorithm. For example, mathematicians use the method of moving a decimal point not just for converting repeating decimals to fractions, but also in many other cases.

Hi @The-Darkin-Blade!
Nice to hear from you again!
The area of circle is a veeery big topic for a small forum post. There are lots of articles about the formula of a circle's area and other things related to this topic! Hundreds! Even thousands!
Your confusion is normal
To learn more about it, you can start from here, or find more advanced information here.

You can also imagine that a circle is a regular polygon with lots and lots of vertices (actually, infinitely many vertices). To find the area of a regular polygon, you cut it into isosceles triangles (see picture) and then sum up their areas. So the area of a polygon is equal to \(n\times(\frac{1}{2}ah)=\frac{h}{2}\times(na),\) where \(n\) - number of vertices, \(a\) - side length and \(h\) - distance from the center of polygon to its side. The term \(na\) is the perimeter of the polygon. As the polygon becomes more and more like a circle, this value approaches the value of the circle's circumference, which is \(2\pi r,\) and value of \(h\) approaches the circle radius. So, substituting \(2\pi r\) instead of \(na,\) we get: \(\text{area of circle}=\frac{r}{2}\times(2\pi r)=\pi r^2.\)

Studying this in more detail and discovering lots of new and very interesting things will be easier as you get older and study higher-level topics like functions, derivatives, integrals, and many other things.

@Kevin-Li We will get back to you with this... we're thinking about it!