G
Topic
First Poster
Last Poster
Some Important Problems in Mathematics
fossasor   11
N Tuesday at 3:20 PM by fossasor
1
2
3
4
5
6
7
11 replies
fossasor
Apr 1, 2025
fossasor
Tuesday at 3:20 PM
problemo
hashbrown2009   1
N Mar 30, 2025 by Dream9
if x/(3^3+4^3) + y/(3^3+6^3) =1

and

x/(5^3+4^3) + y/(5^3+6^3) =1

find the 2 values of x and y.
1 reply
hashbrown2009
Mar 30, 2025
Dream9
Mar 30, 2025
interesting way to derive the quadratic formula
Soupboy0   2
N Mar 28, 2025 by Mathematicalprodigy37
If you have the quadratic $ax^2+bx+c$, call the roots $r$ and $s$ with $r \ge s$. Then by Vieta's, $r+s = \frac{-b}{a}$ and $rs = \frac{c}{a}$. If we can find $r+s$, we can find $r-s$. Note that $(r-s)^2=r^2-2rs+s^2$ and $(r+s)^2=r^2+2rs+s^2$. Therefore, $(r+s)^2-4rs=(r-s)^2$, and $r-s=\sqrt{(r+s)^2-4rs} = \sqrt{\frac{-b}{a^2}^2-\frac{4c}{a}}=\sqrt{\frac{b^2-4ac}{a^2}}=\frac{\sqrt{b^2-4ac}}{a}$. Now all we have to do is solve the following system:
$r+s=\frac{-b}{a}$ and $r-s=\frac{\sqrt{b^2-4ac}}{a}$. Solving for $r$ by elimination, we get $2r = \frac{-b+\sqrt{b^2-4ac}}{a}$, so ${r=\frac{-b+\sqrt{b^2-4ac}}{2a}}$, which is the quadratic formula. Substituting this into our first equation, we get $\frac{-b+\sqrt{b^2-4ac}}{2a}+s=\frac{-b}{a}$, and solving for $s$ yields $s=\frac{-b-\sqrt{b^2-4ac}}{2a}$,
2 replies
Soupboy0
Mar 28, 2025
Mathematicalprodigy37
Mar 28, 2025
quadratics
luciazhu1105   24
N Mar 23, 2025 by cheltstudent
I really need help on quadratics and I don't know why I also kinda need a bit of help on graphing functions and finding the domain and range of them.
24 replies
luciazhu1105
Feb 14, 2025
cheltstudent
Mar 23, 2025
Three variable equations
gpen1000   1
N Mar 21, 2025 by fruitmonster97
1. For integers $x$, $y$, and $z$ such that $\frac{\sqrt{x}}{\sqrt{y}} = z$, find $\frac{\sqrt{z}}{\sqrt{x}}$ in terms of $x$, $y$, and $z$.

2. For integers $x$, $y$, and $z$ such that $x = y - 1$ and $z = y + 1$, prove that $y^3 = xyz + y$.
1 reply
gpen1000
Mar 21, 2025
fruitmonster97
Mar 21, 2025
Math Question
somerandomkid32   2
N Mar 19, 2025 by Dream9
I was looking to get better at math overall but don't know where to start. For context I am taking geometry as an 8th grader and have gotten a 18 on AMC 8. I have some background in Algebra 2 already such as factoring polynomials etc. reply if you need more info.
2 replies
somerandomkid32
Mar 19, 2025
Dream9
Mar 19, 2025
2014 National MathCounts Problem 28
ilikemath247365   5
N Mar 15, 2025 by ilikemath247365
If $f(x) = \frac{ax + b}{cx + d}, {abcd \neq 0}$ and $f(f(x)) = x$ for all $x$ in the domain of $f$, what is the value of $a + d$?
5 replies
ilikemath247365
Mar 15, 2025
ilikemath247365
Mar 15, 2025
2000th post!
evt917   35
N Mar 10, 2025 by eddie.li
Wow I can't believe I'm at 2000 posts already! I guess this also celebrates my (late) 3 year anniversary on AoPS!

um i guess i share my story (most problems are written by me)

1st grade -- i forgot ok

2nd grade -- i was at public school like the regular kids, there i started loving basketball and i was already working on prealgebra level stuff example problem

3rd grade -- started aops i skipped to calculus anyway so here i was still in public school and I started algebra a, but made no progress so my parents asked me if I want to homeschool (lucky they didn't force me), and i said yes example problem

4th grade -- Finished intro to programming with python, got 13 on amc 8 (skull), finished all intro courses except intro to number theory. i was shaky on lots of concepts and i had to do a review (by myself with the aops books) sometimes. example problem

5th grade -- finished intermediate programming with python but somehow failed usaco (the score shall be undisclosed), i started learning some basic C++, and finished all intro courses and im doing intermediate algebra and intermediate number theory now :D . I got 21 on amc 8 (improvement but no dhr) and 72 on amc 10a (buh). oh and by the way I'm still playing basketball in my rec league :) example problem

for more example problems go to my two mock amc 8's and keep an eye on my other mock im about to create!

anyway thanks aops for 2000 posts it helped me learn so much


p.s. lots of information i didn't share but this is the general idea (also pls upvote! if i reach 10 upvotes i will create something special here)
35 replies
evt917
Feb 24, 2025
eddie.li
Mar 10, 2025
Min value
Mathsboy100   2
N Mar 1, 2025 by theclockhasstruck
If $|2x-3|\leq5$ and $|5-2y|\leq3$ find the least possible value of $x-y$
2 replies
Mathsboy100
Feb 26, 2025
theclockhasstruck
Mar 1, 2025
Interesting Factoring Result
paganiniana   10
N Mar 1, 2025 by theclockhasstruck
Factor $b^5+b+1$.

Hint
10 replies
paganiniana
Jul 5, 2024
theclockhasstruck
Mar 1, 2025
Questions about dividing by 0
Arr0w   28
N Nov 15, 2024 by b2025tyx
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
1
2
3
4
5
28 replies
Arr0w
Dec 2, 2020
b2025tyx
Nov 15, 2024
Questions about dividing by 0
G H J
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Arr0w
2908 posts
#1 • 2 Y
Y by mobro, DDCN_2011
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
1
2
3
4
5
This post has been edited 1 time. Last edited by Arr0w, Dec 2, 2020, 5:02 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Math2k06
148 posts
#2 • 3 Y
Y by Mango247, Mango247, Mango247
$1/0$ is not a number.For #3, since anything times $0$ is $0$, the answer should be $0$.

Tldr don't poke into questions including $0$ and division. You will open a parallel univeerse
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
aaja3427
1918 posts
#3
Y by
S2

@above that wouldn't be true since you are multiplying by undefined. Anything to do with undefined is undefined.
This post has been edited 1 time. Last edited by aaja3427, Dec 2, 2020, 5:08 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mop
4053 posts
#4
Y by
Undefined numbers are a class of their own. Thus, they cannot be used with real or imaginary numbers without creating an undefined result.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
cubingsoda
19221 posts
#5 • 3 Y
Y by Mango247, Mango247, Mango247
see this video

He divides $a - b$ but since $a=b$ he divides by $0$. He gets $1=2$!

Dividing by $0$ opens the black hole
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
correcthorsebatterystaple
620 posts
#6
Y by
Yeah, there are a couple of constructs in higher maths which assign a value to 1/0 (sort of like $i$) but you have to bend the rules of arithmetic a bit and what's going on here (1-3) is definitely not that. They're not that useful, either. We leave 1/0 undefined most of the time, similar to how you would say "no solutions" if you were asked to solve $x^2=-1$ over $\mathbb{R}.$ (to answer 5)

An "undefined value" has no numerical value; I think even calling it a "value" in the first place is a bit of a misnomer.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ihatemath123
3441 posts
#7
Y by
S4
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
cubingsoda
19221 posts
#8
Y by
undefined = no answer so no solution
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SteindorfStrongGirl
116 posts
#9
Y by
i made a presentation on this at school :< its something about 1≠2
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Awesome_Twin1
706 posts
#12
Y by
1. Avoid double posting
2. Don't bump this thread. Just look at A Letter to MSM which has been pinned.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
A_MatheMagician
2251 posts
#13
Y by
please read this post first
sry @above sniped me
This post has been edited 1 time. Last edited by A_MatheMagician, Dec 14, 2023, 1:45 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
vrondoS
163 posts
#14
Y by
In some ways, you can think $\frac{1}{0}=\infty$. This isn't rigorous, however, because then $\frac{2}{0}=2\infty=\infty$. Thinking about it this way can help explain some of the questions you had.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
A_MatheMagician
2251 posts
#16
Y by
Post #14 by vrondoS
infinity is not a value
that does not make any sense
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
ZekeMath
41 posts
#17
Y by
If it's undefined, eventually somebody will define it :) I say better now than later.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Yummo
295 posts
#18
Y by
Awesome_Twin1 wrote:
1. Avoid double posting
2. Don't bump this thread. Just look at A Letter to MSM which has been pinned.
A_MatheMagician wrote:
please read this post first
sry @above sniped me

Do you realize who posted that? @Arr0w, I thought you left AoPS a while ago.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
mathboy282
2989 posts
#20
Y by
vrondoS wrote:
In some ways, you can think $\frac{1}{0}=\infty$. This isn't rigorous, however, because then $\frac{2}{0}=2\infty=\infty$. Thinking about it this way can help explain some of the questions you had.

I would argue that this isn't true. $1/0 is undefined.$ The limit of it also does not exist, because:
$\lim_{x->0^+}\frac1x = +\infty$
but also:
$\lim_{x->0^-} \frac1x = -\infty$
This post has been edited 2 times. Last edited by mathboy282, Dec 14, 2023, 3:33 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
JoyfulSapling
822 posts
#21
Y by
Arr0w wrote:
I have a couple of questions all of which have to do with dividing by 0. Thanks in advance.
1
2
3
4
5

Solutions:
1
2
3
4
5
This post has been edited 4 times. Last edited by JoyfulSapling, Mar 14, 2024, 2:33 PM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
MathPerson12321
3639 posts
#22
Y by
I would think a variable like b would be defined as $\frac{1}{0}$, similar to $i$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
the_mathmagician
467 posts
#23
Y by
Note: defining a number like $b=\frac{1}{0}$ serves no purpose. Why did we define $i$? Because we created the quadratic formula and realized that we couldn't describe the full range of solutions. Plus, we found something called casus irreducibilis, in which we found that describing a root of a cubic that was a real number required the use of imaginary numbers. After we accepted that we found a lot of useful other things to do with complex numbers. As for with "$b$", it serves no purpose. What can we do with this? There's nothing we can do with it. Actually, all it does is completely break our current framework of math.

Edit: Also, read this by the OP, written a few years later. It's an announcement but MSM unfortunately isn't known for paying attention to those ;)
This post has been edited 2 times. Last edited by the_mathmagician, Mar 14, 2024, 4:10 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
yummy-yum10-2021
95 posts
#24
Y by
Arr0w wrote:
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ 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.
[...]
What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.
[...]
What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[...]
INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ 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 $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, 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
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]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 $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.


Full post: A Letter to MSN
This post has been edited 1 time. Last edited by yummy-yum10-2021, Nov 11, 2024, 3:41 AM
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
greenplanet2050
1291 posts
#25
Y by
yummy-yum10-2021 wrote:
Arr0w wrote:
Greetings.

I have seen many posts talking about commonly asked questions, such as finding the value of $0^0$, $\frac{1}{0}$,$\frac{0}{0}$, $\frac{\infty}{\infty}$, why $0.999...=1$ 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.
[...]
What about $\frac{1}{0}$? Suppose that $x=\frac{1}{0}$. Then we would have $x\cdot 0=0=1$, absurd. A more rigorous treatment of the idea is that $\lim_{x\to0}\frac{1}{x}$ does not exist in the first place, although you will see why in a calculus course. So the point is that $\frac{1}{0}$ is undefined.
[...]
What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[...]
INDETERMINATE VS UNDEFINED

What makes something indeterminate? As you can see above, there are many things that are indeterminate. While definitions might vary slightly, it is the consensus that the following definition holds: A mathematical expression is be said to be indeterminate if it is not definitively or precisely determined. So how does this make, say, something like $0/0$ 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 $0/0$ is an indeterminate form.

But we need to more specific, this is still ambiguous. An indeterminate form is a mathematical expression involving at most two of $0$, $1$ or $\infty$, 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
\[0/0, \infty/\infty, \infty-\infty, \infty \times 0\]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 $f:\mathbb{R}^{+}\cup\{0\}\rightarrow\mathbb{R}$ given by the mapping $x\mapsto \sqrt{x}$ is undefined for $x<0$. On the other hand, $1/0$ is undefined because dividing by $0$ is not defined in arithmetic by definition. In other words, something is undefined when it is not defined in some mathematical context.


Full post: A Letter to MSN
There’s no need to bump this thread.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
vincentwant
1282 posts
#26
Y by
me when the person who posted this is the same person who posted a letter to msm:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Turtle09
1805 posts
#27
Y by
vincentwant wrote:
me when the person who posted this is the same person who posted a letter to msm:

this is crazy lol
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
sadas123
1115 posts
#28
Y by
honestly I think that 1/0 is infinity because if you graph this on a graphing calculator it looks like this and you can see the largest red line is when you divide with 0 and if you do it with a smaller number with 1 which I will also include makes it a very large slope change, which looks like it is going to infinity.
Attachments:
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
club52
724 posts
#29
Y by
@above it goes to both positive and negative infinity.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
b2025tyx
1441 posts
#30
Y by
Sol 4
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
blimpo
801 posts
#31
Y by
I would like to point out that undefined and infinity are not numbers, but definitions. So you can't say "x is equal to infinity" or "when dividing this by undefined"
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Catcumber
159 posts
#32
Y by
guys stop bumping this thread... its 4 years old
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
b2025tyx
1441 posts
#33
Y by
Catcumber wrote:
guys stop bumping this thread... its 4 years old

Whoops, forgot to check that
Z K Y
N Quick Reply
G
H
=
a