Disturbing math facts that you didn't ask for
by greenturtle3141, Dec 19, 2021, 8:34 PM
Reading Difficulty: 0-5/5
Some classics:
Some others:
Feel free to share others in the comments.
Some classics:
of 
is equal to 
of 
.- It's possible to drill a hole into a cube such that you can pass a bigger cube through the hole.
- Take a map of the world. Crumble it into a ball and throw it outside. There exists a point on the map that lies directly above the real-world location in the world that it corresponds to.
- The harmonic series diverges. But it will converge if you remove terms that contain a
as a digit in the denominator.
of 
is equal to 
of 
.
as a digit in the denominator.
be the point in the world that's directly under the point on the map corresponding to the point 
is a continuous map from the world to the world. Some compactness argument lets you apply Brouwer's to deduce that Some others:
- The last digit of
is not 
. 
for all .- Consider a unit cube in
. It can be anywhere and with any orientation. No matter what, the length of its projection unto the 
-axis will be equal to the area of its projection unto the 
-plane. - There exists a relation that is symmetric and transitive, but not reflexive.
- How many open covers are there of
in the empty topology? - Suppose
is a monic polynomial with integer coefficients whose roots are real numbers in 
. Then 
. - Is it true that every chain of subsets of
is countable?
is not 
.
for all 
.
-
-
in the empty topology? 
is a monic polynomial with integer coefficients whose roots are real numbers in 
.
.
is countable? 
such that for any 
,
or 
.
is a chain of subsets of 
.
on 
,
.
both work.
be the roots of 
.![$x \in [0, 3]$](http://latex.artofproblemsolving.com/7/7/a/77ac5e3365959039da8431fe01d976f9d70a5b21.png)
,![\[ |x(x-1)(x-2)(x-3)| \le 1, \]](http://latex.artofproblemsolving.com/1/9/0/190ad09c14bd9974119d576b739b213b492fd5ba.png)
with equality iff 
.![\[ x(x-1)(x-2)(x-3) = (x^2-3x)(x^2-3x+2) = (x^2-3x+1)^2 - 1. \]](http://latex.artofproblemsolving.com/6/f/1/6f1511bb1ca13cbc4bdba1e0f15b6612458b9f06.png)
It is now not hard to finish verifying the inequality and finding its equality cases. 
,![\[ |Q(0)Q(1)Q(2)Q(3)| = \prod_{i=1}^n |x_i(x_i-1)(x_i-2)(x_i-3)| \le 1. \]](http://latex.artofproblemsolving.com/e/a/a/eaa00d4bd4b9f355cce6746773e1d1490dfaf35d.png)
On the other hand, since 
.
for all 
.
.
.
for 
.Feel free to share others in the comments.
This post has been edited 4 times. Last edited by greenturtle3141, Dec 19, 2021, 8:45 PM
and 
then it's not that interesting
is prime for 
but not 
points on a circle and draw all line segments between points so that no three lines intersect at the same point, what's the maximum number of regions that the lines divide the circle into?
you get 
all powers of two.
you get 
regions. The number is 
I first saw this in an exposition by Paul Zeitz in a decade of berkeley math circle in the combinatorics chapter. This problem was how he introduced the chapter lol.
the thing is never a power of two.
unfortunately I haven't much experience with them yet. i only learned about runge's phenomenon because i was naively trying interpolation to prove weierstrass approximation for a long long time... and gave up and read the wikipedia page and found it haha. and it's also mentioned in this excellent article 
so 
for some 
)
there exists some point 
and 
for any smaller 
and 
precedes 
and i was doing some research and found 
staircase "approximation" where convergence fails... though unfortunately this seems to be more of a meme than something disturbing. maybe disturbing if you show your math teacher haha
March 2025
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