,
,
,
,
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 given integer. Prove that infinitely many terms of the sequence 
,![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
,
denotes the largest integer not exceeding 
all of whose digits are odd.
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
.
memorized but I'm sure there are people out there with more!
indicates my favorites.
is defined over the positive integers as follows: 
,
for 
prime, and for all relatively prime positive integers 
and 
,
.
is the smallest positive integer such that 
,
satisfies 
,
,
,
,
,
?
and 
be the greatest odd divisor of 
for even 
for positive integers 
and 
where 
is minimized, find the number of divisors of 
.
consists of a line and a point not on the line. The distance from the point to the line can be expressed as 
,
.
.
are positive real numbers such that 
,
,
,
be a convex pentagon satisfying 
,
,
.
be the intersection of lines 
and 
.
has a perimeter of 
and an area of 
,
be an acute scalene triangle with longest side 
.
be the circumcenter of 
are chosen on 
and 
.
,
,
,
.
such that 
is divisible by 
by 
,
,
,
such that in every row and column, the number of 
or 
is valid, while a row with squares labeled 
or 
is not valid.) Find the remainder when 
.
,
be real numbers satisfying the system of equations
What is the value of 
?
at 
is the reflection of 
and 
are 
.
be the set of all 
,
distributing the modulo operation. This simplifies into 0 + a mod b since a is always divisible by a except for when it is 0 so a diamond b is 
.
and 
in a diamond b to get
and 
in a heart b and distribute the modulo as well
Y by
Let 
be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular 
-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let 
denote the number of different ways Luca can make jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.)
a) Find the value of if is odd.
b) Prove that if is even, then
![\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]](//latex.artofproblemsolving.com/3/0/9/30992e5982bce4a032d5e96e0fb5fce8ab6ff21a.png)
Proposed by Péter Csikvári and Kartal Nagy, Budapest
be a given positive integer. Luca, the lazy flea sits on one of the vertices of a regular 
-gon. For each jump, Luca picks an axis of symmetry of the polygon, and reflects herself on the chosen axis of symmetry. Let 
denote the number of different ways Luca can make jumps such that she returns to her original position in the end, and does not pick the same axis twice. (It is possible that Luca's jump does not change her position, however, it still counts as a jump.)a) Find the value of
if is odd.b) Prove that if
is even, then![\[P(n)=(n-1)!\cdot n!\cdot \sum_{d\mid n}\left(\varphi\left(\frac{n}d\right)\binom{2d}{d}\right).\]](http://latex.artofproblemsolving.com/3/0/9/30992e5982bce4a032d5e96e0fb5fce8ab6ff21a.png)
Proposed by Péter Csikvári and Kartal Nagy, Budapest
