,
,
,
,
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 
,![\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]](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 
such that every collection of 
.
on one side and a 
on the other. Harry has 
of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly 
coins showing 
the process starting with the configuration 
would be 
,
,
be the number of operations before Harry stops. For example, 
and 
.
possible initial configurations 
be the foot of the altitude from the vertex 
in the acute-angled 
.
be the midpoint of side 
,
be the circumcentre of 
meets line 
at 
.
,
board where we identify the squares with pairs 
where 
denote the row and column number of that square, respectively.
and places a pawn at the bottom left corner (i.e. on 
)
or 
if 
and 
if 
that Calvin could have picked such that he can make moves so that the pawn covers all the squares on the board without being on any square twice.
has how many digits?
and 
intersect at the point 
?
,
.
are chosen from 0 to 1. What is the probability that their positive difference is more than 
?
?
families, there are 
children respectively. If a random child from any of the families is chosen, what is the probability that the child has 
siblings? Express your answer as a common fraction.[/quote]
,
people, exactly 
such that 
.
,
such that 
.
.
has 
,
,
.
.
and the distance between 
and 
is 
Point 
such that 
What is 
and 
is the circumcenter of 
If 
what is 
,![\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]](http://latex.artofproblemsolving.com/8/2/f/82ffc15072ca99f6cd1abc312781eb4827817356.png)
Y by RainbowJessa, radian_51, farhad.fritl, dangerousliri
In each cell of a 
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
What is the largest possible value of
?
Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.What is the largest possible value of
?Proposed by Paulius Aleknavičius, Lithuania, and Anghel David Andrei, Romania
This post has been edited 2 times. Last edited by Lukaluce, Apr 29, 2025, 10:37 AM
is 
.
columns into 
squares, and put 
in each of the main diagonals of the squares and 
.
and 
,
.
.
be the number of circles in column 
be the sum of the circles in column 
.
,
as 
is equivalent to 
since each circled number must be at least 
.
rectangle with 
rectangle. Fill the 
,
with ![\[
\frac{R}{C} = \frac{2025 \cdot \frac{1}{45}}{45 \cdot \frac{1}{45} + \frac{2025 - 45}{45}} = \frac{2025}{89}
\]](http://latex.artofproblemsolving.com/c/b/3/cb30fdd6325affe7ccf8b8b9fe3364965db08e44.png)
remains the same and 
.
.
.![\[
a_1r_1 + a_2r_2 + \dots + a_ir_i \le N \cdot \left(r_1 + r_2 + r_3 + \dots + \frac{2025 - i}{2025}\right)
\]](http://latex.artofproblemsolving.com/1/d/5/1d586dac5b2dd959c42ceecd6f13cccf31a0d9d2.png)
This is tightest when the 
have 
and the 
have 
.![\[
j + \frac{t}{2025} \le N \cdot \left(\frac{1}{a_1} + \dots + \frac{1}{a_j} + \frac{i-j}{2025} + \frac{2025 - i}{2025}\right)
\]](http://latex.artofproblemsolving.com/1/3/4/134b1b2312664e85d716e7095387c4c850265866.png)
where 
.
,![\[
\frac{j}{N} \le \frac{1}{a_1} + \dots + \frac{1}{a_j} + \frac{2025 - i}{2025}
\]](http://latex.artofproblemsolving.com/c/c/b/ccb98feb86ff336cda077524517d61d7338a5960.png)
which with AM-HM and applying 
becomes![\[
\frac{j}{N} \le \frac{j^2}{a+j} + \frac{a}{2025}
\]](http://latex.artofproblemsolving.com/e/a/1/ea1ea3b356d56851573e1e7569ee2debe1e97438.png)
with 
.
as the only root. If 
,![\[
\frac{j}{N} \le \frac{j^2}{45j} + \frac{44j}{2025} \iff
\frac{89j}{2025} \le \frac{45j}{2025j} + \frac{44j}{2025}
\]](http://latex.artofproblemsolving.com/9/e/b/9eb5abf9121eeae979999aa3df959f986e75b35d.png)
which holds. If 
,![\[
\frac{j}{N} \le \frac{j^2}{2025} + \frac{2025-j}{2025} \iff
89j \le j^2 + 2025 - j \iff (j-45)^2 \ge 0
\]](http://latex.artofproblemsolving.com/7/0/9/709b85997c99d6f948062143604dd238d5b5bc8c.png)
which holds.
.
,
(everything should be divided by 
)
columns that contains scorers. Note that the non-scorers in a column with more than 
scorers should be distributed to the scorers to maximize the ratio, the converse is true (distribute among non-scorers if there aren't enough scorers).![\[\frac{n^2a+n^2-s}{n^2-a+\frac{n^2a^2}{s}}\le\frac{n^2}{2n-1}.\]](http://latex.artofproblemsolving.com/c/6/e/c6ecd4d4270253776f6af21ed254ba252c09f0d1.png)
After more bashing, we realize that this is optimized when 
,
the number of yellow cells it has.
and all we want is for this to be in terms of 
is trivially true by double counting so now we will split sum of 
and therefore 
And now here we need 
and thus 
but also trivially 
which should lead to picking 
for some 
,
and therefore 
so our pick here will be 
to get rid of each 
in which case we get that 
this so that we don't have the need to summon each 
so tracing back the equality this gives that in each of the columns where 
we have that the only nonzero cells are the ones in yellow, also from the AM-GM we have that 
if 
'
.
.
,
in each such column, we guarantee that each heavy column carries the minimal possible maximum to serve 45 rows. (A short optimization shows that if one designates 
columns to serve as the targets for high entries and distributes the mass evenly among them so that every heavy row gets exactly 
in the column it uses, then one may obtain a ratio of![\[ \frac{n}{\,k+r-1}\,, \]](http://latex.artofproblemsolving.com/b/5/c/b5c9448f65ff9408fbdbb06068b0f8da03f56b39.png)
with the constraint 
.
;
,![\[ \frac{2025}{45+45-1}=\frac{2025}{89}\,. \]](http://latex.artofproblemsolving.com/d/d/8/dd8e174440dcffbb58f3bc3039935de45724c965.png)
)
block with the “flat” matrix on the remaining cells.)
rows into 
.
.
numbers in those columns and the sum in each row is 
.
because they are contained in 
.
'
.
,
.
which will get us to 
which yields the needed answer.
.
:![\[\begin{tabular}{c|c|c|c} $1/2$ & 0 & $1/4$ & $1/4$ \\ \hline
$1/2$ & 0 & $1/4$ & $1/4$ \\ \hline
0 & $1/2$ & $1/4$ & $1/4$ \\ \hline
0 & $1/2$ & $1/4$ & $1/4$ \\
\end{tabular}\]](http://latex.artofproblemsolving.com/0/e/8/0e8d2b6de9d484adf0fdbe0f2823b42bfc0c8af5.png)
In general, for an 
grid, we place fractions 
in groups of 
columns with 
.
distinct columns (without loss of generality columns 
)
,
The first inequality follows by AM-GM, and the last follows by considering a quadratic in 
.![\[\frac RC \leq \frac{\sum_{i=1}^d s_i}{\left(\frac 2{\sqrt n} - \frac 1n\right) \sum_{i=1}^d s_i} = \left(\frac 2{\sqrt n} -\frac 1n\right)^{-1} = \frac{2025}{89}\]](http://latex.artofproblemsolving.com/c/b/1/cb1ffc93934125d4ee0d817e7cd85e6c0432b506.png)
when 
.
