AoPS Academy
, 
,
, 
, why 
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.. It is usually regarded that 
, not because this works numerically but because it is convenient to define it this way. You will see the convenience of defining other undefined things later on in this post.? The issue here is that 
isn't even rigorously defined in this expression. What exactly do we mean by ? Unless the example in question is put in context in a formal manner, then we say that is meaningless.? Suppose that 
. Then we would have 
, absurd. A more rigorous treatment of the idea is that 
does not exist in the first place, although you will see why in a calculus course. So the point is that is undefined.
? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
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 is an indeterminate form.
, 
or , obtained by applying the algebraic limit theorem (a theorem in analysis, look this up for details) in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being calculated. This is why it is called indeterminate. Some examples of indeterminate forms are![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](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 
. On the other hand, 
is undefined because dividing by is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context. is considered both indeterminate and undefined (but in the context of a limit then it is considered in indeterminate form). Additionally, this notion of something being undefined also means that we can define it in some way. To rephrase, this means that technically, we can make something that is undefined to something that is defined as long as we define it. I'll show you what I mean.![\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]](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 it is perfectly OK to say that,![\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 is undefined still! Is there a place where it isn't? Kind of. To do this, we can extend the complex numbers! More formally, we can define this extension as![\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]](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 is allowed here! In fact, we have![\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]](http://latex.artofproblemsolving.com/1/6/0/160cb939707708ff4e0930724df2928af11d7fd1.png)
where 
and are left undefined. We also have
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 , by defining them as![\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]](http://latex.artofproblemsolving.com/c/0/4/c044ff37849acad3b8b280b9753539225a1276ca.png)
They behave in a similar way to the Riemann Sphere, with division by also being allowed with the same indeterminate forms (in addition to some other ones).
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
be an acute triangle. Let 
be any point on the side 
. On line 
, choose any point 
. Let 
and 
be the tangents from to the circumcircle of 
, and let 
and 
be the tangents from to the circumcircle of 
. Find the locus of the intersection of 
and 
.
, 
is bisector of 
and 
. and are feet of perpendicular from 
to and 
,respectively.Prove that orthocenter of triangle 
is on 
.
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 
, 
. If 
is the smallest positive integer such that 
, find the units digit of .
If convex quadrilateral satisfies 
, 
, 
, 
, and 
, what is the value of 
? Express your answer in simplest radical form.
and 
be the greatest odd divisor of . Let 
for even 
denote the product of every odd positive integer less than . If 
for positive integers 
and 
where 
is minimized, find the number of divisors of 
. There exists exactly one positive real number such that the graph of the equation 
consists of a line and a point not on the line. The distance from the point to the line can be expressed as 
, where and are positive integers and is not divisible by any square greater than . Find 
.
. If 
are positive real numbers such that 
, 
, and 
, find . Express your answer as a common fraction. Let 
be a convex pentagon satisfying 
, 
, 
. Let be the intersection of lines 
and . If 
has a perimeter of 
and an area of 
, find the area of .
Let 
be an acute scalene triangle with longest side . Let 
be the circumcenter of . Points and are chosen on such that 
and 
. If 
, 
, and 
, the area of the circumcircle of can be expressed as 
. Find .
such that 
is divisible by .
Alex has a 
by grid of squares. Let be the number of ways that Alex can fill out each square with one of the letters , 
, 
, or such that in every row and column, the number of 's and 's are the same, and the number of 's and 's are the same. (For example, a row with squares labeled 
or 
is valid, while a row with squares labeled 
or 
is not valid.) Find the remainder when is divided by 
.
by grid of squares along the gridlines into two or more pieces such that if three pieces meet at a point , then there are actually four pieces with a vertex at ? An example is shown below.
be real numbers satisfying the system of equations
What is the value of 
? In , the incircle is tangent to 
at , and 
is the reflection of across the midpoint of . Suppose that the inradii of 
and 
are and respectively, and the distance between their incenters is 
. What is the inradius of ? Express your answer as a common fraction. be a positive integer and let 
be the set of all -tuples of 's and 's. Two elements of are said to be neighboring if and only if they differ in only one coordinate. Bob colors the elements of red and blue such that each blue -tuple is neighboring to exactly two red -tuples and no two red -tuples neighbor each other. If 
, find the least possible value of .
be the remainder when![\[100x^{100} + 99x^{99} + 98x^{98} + \cdots + 3x^3 + 2x^2 + x\]](http://latex.artofproblemsolving.com/d/5/4/d5401c220a8682335a1306ae790a80947a38d3f6.png)
is divided by 
Solve for 
![\[
\frac{4000^2+4050^2}{(2000+2025)^2+(2000-2025)^2}
\]](http://latex.artofproblemsolving.com/b/9/9/b991bda876c059439578dd1eeb54b1f8010df755.png)
, 
, 
, and 
be the orthocenters of triangles 
, 
, 
, and 
, respectively. Let 
be the centroid of triangle . is the foot of onto the plane .
be the foot of onto the plane . Notice that 
is perpendicular to plane 
. Thus 
, which implies 
. Similarly, 
. Thus 
, as required. and . Notice the centroids of triangles , , and all coincide at some point . Since is the foot of onto the plane , it follows that 
. By the claim, we also get that 
. Thus all five points lie on the sphere with diameter 
, concluding the proof.
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
Therefore, 
![\[R(0) = 100(0^{100}) + 99(0^{99}) + 98(0^{98}) + \cdots + 3(0^{3}) + 2(0^{2}) + 0\]](http://latex.artofproblemsolving.com/0/0/4/0044c0eabc8eb4ba3d6b430ab908b3e92ba242f1.png)
![\[a(0)^2 + b(0) + c = 0\]](http://latex.artofproblemsolving.com/1/2/4/124caf8e3542d5c7e7d4b13b8324ad59758e5b8a.png)
![\[c=0\]](http://latex.artofproblemsolving.com/3/e/8/3e87824860754458838fc46dd729d700589ccbdb.png)
Substitute 
![\[R(1) = 100(1^{100}) + 99(1^{99}) + 98(1^{98}) + \cdots + 3(1^{3}) + 2(1^{2}) + 1\]](http://latex.artofproblemsolving.com/9/a/1/9a1757b781ed32cb6679971426b2ab28ecade5c7.png)
![\[a+b+c = \frac{(100)(101)}{2}\]](http://latex.artofproblemsolving.com/0/6/d/06d75968cb318133118fac2eac69df3e0c956e93.png)
![\[\boxed{a+b = 5050}\]](http://latex.artofproblemsolving.com/8/7/7/877ee0cf9be48f8dcee17815c0bd35deb3d3edb5.png)
Substitute 
![\[R(-1) = 100(-1^{100}) + 99(-1^{99}) + 98(-1^{98}) + \cdots + 3(-1^{3}) + 2(-1^{2}) - 1\]](http://latex.artofproblemsolving.com/0/7/5/075d546e826059f361650e03e721c5c148f65839.png)
![\[a - b + c = 100-99+98 \cdots -3 + 2 - 1\]](http://latex.artofproblemsolving.com/c/c/a/cca420756d22a06d25afe74d37d55465258c4888.png)
![\[\boxed{a-b = 50}\]](http://latex.artofproblemsolving.com/b/2/9/b292e0f9e4c3f42e823db2d4365bb991f5d172db.png)
Hence, 
and 
so 
is given. It is known that triangles 
,
,
,
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle 