,
,
,
,
or even expressions of those terms combined as if that would make them defined. I have made this post to answer these questions once and for all, and I politely ask everyone to link this post to threads that are talking about this issue.
,
isn't even rigorously defined in this expression. What exactly do we mean by 
.
,
does not exist in the first place, although you will see why in a calculus course. So the point is that 
?
indeterminate? In analysis (the theory behind calculus and beyond), limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits. However, if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an indeterminate form. For example, we could say that 
,
or ![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](http://latex.artofproblemsolving.com/2/6/0/260f97a1cf1ea8722ff97a29b83bac7bfe83e577.png)
etc etc. So what makes something undefined? In the broader scope, something being undefined refers to an expression which is not assigned an interpretation or a value. A function is said to be undefined for points outside its domain. For example, the function 
given by the mapping 
is undefined for 
.
is undefined because dividing by ![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](http://latex.artofproblemsolving.com/9/9/3/993c78b62ec2c84b5d77daa8e9a8cd6f70fcf904.png)
So instead of treating infinity as an idea, we define infinity (positively and negatively, mind you) as actual numbers in the reals. The advantage of doing this is for two reasons. The first is because we can turn this thing into a totally ordered set. Specifically, we can let 
for each 
which means that via this order topology each subset has an infimum and supremum and 
is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In ![\begin{align*}
a + \infty = \infty + a & = \infty, & a & \neq -\infty \\
a - \infty = -\infty + a & = -\infty, & a & \neq \infty \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \pm\infty, & a & \in (0, +\infty] \\
a \cdot (\pm\infty) = \pm\infty \cdot a & = \mp\infty, & a & \in [-\infty, 0) \\
\frac{a}{\pm\infty} & = 0, & a & \in \mathbb{R} \\
\frac{\pm\infty}{a} & = \pm\infty, & a & \in (0, +\infty) \\
\frac{\pm\infty}{a} & = \mp\infty, & a & \in (-\infty, 0).
\end{align*}](http://latex.artofproblemsolving.com/6/d/8/6d8fc1eca5c791d430a7168cd3b37a02104fa318.png)
So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,![\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]](http://latex.artofproblemsolving.com/a/0/b/a0b67076efcc014c78b2023be1151c84b57c1503.png)
So while we define certain things, we also left others undefined/indeterminate in the process! However, in the context of measure theory it is common to define 
as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by ![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](http://latex.artofproblemsolving.com/8/4/e/84ed3d6ab237df970333b87beaef5311caa46185.png)
which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, 
means complex infinity, since we are in the complex plane now. Here's the catch: division by ![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and 
Furthermore, we actually have some nice properties with multiplication that we didn't have before. In 
it holds that![\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]](http://latex.artofproblemsolving.com/8/e/0/8e035b4841d20c5ad671c46355cc667ca01e533b.png)
but 
and 
are left as undefined (unless there is an explicit need to change that somehow). One could define the projectively extended reals as we did with ![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by 
be a real numbers such that 
T
and 
Prove that
segments 
and 
are parallel. Angle bisectors of 
and 
meet at 
.
and 
meet at 
.
is cyclic
is even , 
is odd and 
is divisible by 
.Prove that there exists a prime number 
such that 
different icons.
are
Y by dangerousliri, megarnie
An integer 
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
is called 
if there exists a permutation 
of the numbers 
, such that:
and 
have different parities for every 
;
the sum 
is a quadratic residue modulo 
for every 
.Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
This post has been edited 2 times. Last edited by swynca, Apr 27, 2025, 4:15 PM
works.
,
into 
pairs 
such that 
,
and 
.
,
is odd and, for all 
,
.
this will be 
,
will be either 
,
.
in the permutation.
such that 
and 
.
,
,
,
,
.
solves this, as Double07 showed in his/her post.
,
.
,
.
assures that no such 
exists.
.
so that they consist of all the even quadratic residues modulo 
.
so that they consist of all the odd quadratic residues modulo 
,
.
.
.
must also be 
,
mod 
are good
pk for some integer k
mod 
as the odd numbers which are QR mod p and lesser than p. . Here
where r is the amount of odd numbers which are QR mod p and lesser than p.
as the even numbers which are QR mod p and lesser than p. . Here
where k is the amount of even numbers which are QR mod p and lesser than p.
have different parity so do 
in the following sequence which will also satisfy the other conditions clearly as at any point it will be either 0
,
,
,
,
,
,
,
,
,
where K is odd are not good (They are bad)
mod 4
and 
are not QRs mod 4
mod 4 for all 
but that is nonsense as for all 
there is an 
.
is a good number where 
.
is a complete residue for 
modulo 
.
there exist a unique 
such that 
and we have that 
have same parities. Now to finish the problem we use the identity,
is NQR mod 
which is an NQR by multiplicativiness of the Legendre Symbol and therefore we can split in 
for which 
and one is a QR and the other is NQR, and the idea for the parity part is just to put 
and then another odd one and so on until you run out of odd ones then add zero and add even ones and then their pair to finish.
large and the reason for this pick is that clearly we can't have two QRs that are consecutive by 
but at some point we have to add 
is NQR, and vice versa. Then we start pairing the terms as follows:
be all quadratic residues mod 
.
,
where 
is odd, till we run out of odd QR's.
for an odd prime 
