,
,
,
,
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 real numbers such that 
.
and 
.
Prove that
,
be a point on arc 
of circle 
and 
.
and 
be the reflection of 
and 
,
intersects 
,
intersects 
.
lies on 
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
!
candy bars. Let N & M be positive integers, what is N+M?
.
or 
.
and 
so 
This should be final answer. (Mathdash rating 800)
Y by
Let 
and 
be positive integers with 
. In a group of people, each one or always
speak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set
, what is the parity of people who speak to
true? ”, where is a subset of size 
of the set of people. The answer can only
be “
” or “
”.
a) For which values of and is it possible to determine which people speak the truth and
which people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
and 
be positive integers with 
. In a group of people, each one or alwaysspeak the truth or always lie. Arnaldo can ask questions for any of these people
provided these questions are of the type: “In set
, what is the parity of people who speak totrue? ”, where
is a subset of size 
of the set of people. The answer can onlybe “
” or “
”.a) For which values of
and is it possible to determine which people speak the truth andwhich people always lie?
b) What is the minimum number of questions required to determine which people
speak the truth and which people always lie, when that number is finite?
This post has been edited 14 times. Last edited by betongblander, Jun 7, 2021, 8:08 PM
,
about the set ![\[ (p+1) + \sum_{a \in A} \pmod 2. \]](http://latex.artofproblemsolving.com/a/a/0/aa06fdf63273fd2e1568549f5e92de569dc526b2.png)
So in other words, a query amounts to sampling a set of either 
elements 
)
elements 
)
changes no responses.
for 
By summing, one gets the value of 
,
for 
.
for 
.
for 
.
possible final answers for Arnaldo, and each query has two possible responses.
questions, by Pidgeonhole there exists some string of answers that corresponds to more than one possible configuration, hence he cannot determine the configuration, which shows the lower bound.
is even, these two numbers are either both odd or both even. If they are both odd, then the people that said odd are truth-tellers and the people that said even are liars, and vice versa for the both even case. In both cases, he can determine the type of each person in 
people to determine their type, hence done.
the process is impossible. For even 
'
'
people remaining, from which 
queries is sufficient to determine who are liars and truthtellers--by taking a set of 
people reply ``odd" and 
people reply ``even". Then 
possibilities for the distribution of liars.
for each 
answer sequences. Since 
there exists atleast two possible states for some answer sequence for 
denote our required (with ![\[\boxed{f(n,k) = \begin{cases} \infty \text{ if $k$ is odd} \\ n \text{ if $k$ is even} \end{cases}}\]](http://latex.artofproblemsolving.com/8/8/d/88db4b91b2b8d6bc02978e7bf4c06c986688c2dd.png)
To see why 
questions atleast.
