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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Yesterday at 2:14 PM
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Yesterday at 2:14 PM
0 replies
k i A Letter to MSM
Arr0w   23
N Sep 19, 2022 by scannose
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.
[list]
[*]Firstly, the case of $0^0$. It is usually regarded that $0^0=1$, 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.

[*]What about $\frac{\infty}{\infty}$? The issue here is that $\infty$ isn't even rigorously defined in this expression. What exactly do we mean by $\infty$? Unless the example in question is put in context in a formal manner, then we say that $\frac{\infty}{\infty}$ is meaningless.

[*]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 if $0.99999...=1$? An article from brilliant has a good explanation. Alternatively, you can just use a geometric series. Notice that
\begin{align*}
\sum_{n=1}^{\infty} \frac{9}{10^n}&=9\sum_{n=1}^{\infty}\frac{1}{10^n}=9\sum_{n=1}^{\infty}\biggr(\frac{1}{10}\biggr)^n=9\biggr(\frac{\frac{1}{10}}{1-\frac{1}{10}}\biggr)=9\biggr(\frac{\frac{1}{10}}{\frac{9}{10}}\biggr)=9\biggr(\frac{1}{9}\biggr)=\boxed{1}
\end{align*}
[*]What about $\frac{0}{0}$? Usually this is considered to be an indeterminate form, but I would also wager that this is also undefined.
[/list]
Hopefully all of these issues and their corollaries are finally put to rest. Cheers.

2nd EDIT (6/14/22): Since I originally posted this, it has since blown up so I will try to add additional information per the request of users in the thread below.

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.

WHEN THE WATERS GET MUDDIED

So with this notion of indeterminate and undefined, things get convoluted. First of all, just because something is indeterminate does not mean it is not undefined. For example $0/0$ 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.

One example of making something undefined into something defined is the extended real number line, which we define as
\[\overline{\mathbb{R}}=\mathbb{R}\cup \{-\infty,+\infty\}.\]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 $-\infty\le a\le \infty$ for each $a\in\overline{\mathbb{R}}$ which means that via this order topology each subset has an infimum and supremum and $\overline{\mathbb{R}}$ is therefore compact. While this is nice from an analytic standpoint, extending the reals in this way can allow for interesting arithmetic! In $\overline{\mathbb{R}}$ 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*}So addition, multiplication, and division are all defined nicely. However, notice that we have some indeterminate forms here which are also undefined,
\[\infty-\infty,\frac{\pm\infty}{\pm\infty},\frac{\pm\infty}{0},0\cdot \pm\infty.\]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 $\infty \times 0=0$ as greenturtle3141 noted below. I encourage to reread what he wrote, it's great stuff! As you may notice, though, dividing by $0$ 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
\[\mathbb{C}^*=\mathbb{C}\cup\{\tilde{\infty}\}\]which we call the Riemann Sphere (it actually forms a sphere, pretty cool right?). As a note, $\tilde{\infty}$ means complex infinity, since we are in the complex plane now. Here's the catch: division by $0$ is allowed here! In fact, we have
\[\frac{z}{0}=\tilde{\infty},\frac{z}{\tilde{\infty}}=0.\]where $\tilde{\infty}/\tilde{\infty}$ and $0/0$ are left undefined. We also have
\begin{align*}
z+\tilde{\infty}=\tilde{\infty}, \forall z\ne -\infty\\
z\times \tilde{\infty}=\tilde{\infty}, \forall z\ne 0
\end{align*}Furthermore, we actually have some nice properties with multiplication that we didn't have before. In $\mathbb{C}^*$ it holds that
\[\tilde{\infty}\times \tilde{\infty}=\tilde{\infty}\]but $\tilde{\infty}-\tilde{\infty}$ and $0\times \tilde{\infty}$ 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 $\mathbb{C}^*$, by defining them as
\[{\widehat {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}.\]They behave in a similar way to the Riemann Sphere, with division by $0$ also being allowed with the same indeterminate forms (in addition to some other ones).
23 replies
Arr0w
Feb 11, 2022
scannose
Sep 19, 2022
k i Marathon Threads
LauraZed   0
Jul 2, 2019
Due to excessive spam and inappropriate posts, we have locked the Prealgebra and Beginning Algebra threads.

We will either unlock these threads once we've cleaned them up or start new ones, but for now, do not start new marathon threads for these subjects. Any new marathon threads started while this announcement is up will be immediately deleted.
0 replies
LauraZed
Jul 2, 2019
0 replies
k i Basic Forum Rules and Info (Read before posting)
jellymoop   368
N May 16, 2018 by harry1234
f (Reminder: Do not post Alcumus or class homework questions on this forum. Instructions below.) f
Welcome to the Middle School Math Forum! Please take a moment to familiarize yourself with the rules.

Overview:
[list]
[*] When you're posting a new topic with a math problem, give the topic a detailed title that includes the subject of the problem (not just "easy problem" or "nice problem")
[*] Stay on topic and be courteous.
[*] Hide solutions!
[*] If you see an inappropriate post in this forum, simply report the post and a moderator will deal with it. Don't make your own post telling people they're not following the rules - that usually just makes the issue worse.
[*] When you post a question that you need help solving, post what you've attempted so far and not just the question. We are here to learn from each other, not to do your homework. :P
[*] Avoid making posts just to thank someone - you can use the upvote function instead
[*] Don't make a new reply just to repeat yourself or comment on the quality of others' posts; instead, post when you have a new insight or question. You can also edit your post if it's the most recent and you want to add more information.
[*] Avoid bumping old posts.
[*] Use GameBot to post alcumus questions.
[*] If you need general MATHCOUNTS/math competition advice, check out the threads below.
[*] Don't post other users' real names.
[*] Advertisements are not allowed. You can advertise your forum on your profile with a link, on your blog, and on user-created forums that permit forum advertisements.
[/list]

Here are links to more detailed versions of the rules. These are from the older forums, so you can overlook "Classroom math/Competition math only" instructions.
Posting Guidelines
Update on Basic Forum Rules
What belongs on this forum?
How do I write a thorough solution?
How do I get a problem on the contest page?
How do I study for mathcounts?
Mathcounts FAQ and resources
Mathcounts and how to learn

As always, if you have any questions, you can PM me or any of the other Middle School Moderators. Once again, if you see spam, it would help a lot if you filed a report instead of responding :)

Marathons!
Relays might be a better way to describe it, but these threads definitely go the distance! One person starts off by posting a problem, and the next person comes up with a solution and a new problem for another user to solve. Here's some of the frequently active marathons running in this forum:
[list][*]Algebra
[*]Prealgebra
[*]Proofs
[*]Factoring
[*]Geometry
[*]Counting & Probability
[*]Number Theory[/list]
Some of these haven't received attention in a while, but these are the main ones for their respective subjects. Rather than starting a new marathon, please give the existing ones a shot first.

You can also view marathons via the Marathon tag.

Think this list is incomplete or needs changes? Let the mods know and we'll take a look.
368 replies
jellymoop
May 8, 2015
harry1234
May 16, 2018
nth root of a, b in O(sqrt(n))
tenniskidperson3   63
N an hour ago by ihategeo_1969
Source: 2014 USAMO problem 6
Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$, then\[\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.\]
63 replies
+3 w
tenniskidperson3
Apr 30, 2014
ihategeo_1969
an hour ago
Sad Combinatorics II
tastymath75025   26
N 2 hours ago by ray66
Source: 2018 USAMO 6, by Richard Stong
Let $a_n$ be the number of permutations $(x_1, x_2, \dots, x_n)$ of the numbers $(1,2,\dots, n)$ such that the $n$ ratios $\frac{x_k}{k}$ for $1\le k\le n$ are all distinct. Prove that $a_n$ is odd for all $n\ge 1$.

Proposed by Richard Stong
26 replies
1 viewing
tastymath75025
Apr 19, 2018
ray66
2 hours ago
Sad Combinatorics
62861   109
N 2 hours ago by ray66
Source: USAMO 2018 P4 and JMO 2018 P5, by Ankan Bhattacharya
Let $p$ be a prime, and let $a_1, \dots, a_p$ be integers. Show that there exists an integer $k$ such that the numbers
\[a_1 + k, a_2 + 2k, \dots, a_p + pk\]produce at least $\tfrac{1}{2} p$ distinct remainders upon division by $p$.

Proposed by Ankan Bhattacharya
109 replies
1 viewing
62861
Apr 19, 2018
ray66
2 hours ago
Goals for 2025-2026
Airbus320-214   287
N 3 hours ago by ouia1234
Please write down your goal/goals for competitions here for 2025-2026.
287 replies
1 viewing
Airbus320-214
May 11, 2025
ouia1234
3 hours ago
inequality
adamath4ever   0
3 hours ago
$0<a,b,c<1$ real numbers ; such that : $2(a+b+c)+4abc = 3(ab +ac+bc)+1$
Prove that: $a+b+c<3/4$
0 replies
adamath4ever
3 hours ago
0 replies
Private Forum for Geometry Study
fossasor   200
N 4 hours ago by Ravensrule8
Hi all,

I've recently been trying to improve my skill at competition math, but I consistently struggle with geometry. Since doing math is more fun and motivating with others, I've created the GeoPrepClub. This is a private forum to increase geometry skill and help others increase theirs: we'll have problems, marathons, and much more. This is similar to forums such as "AMC8 Prep Buddies" by PatTheKing, but is focused exclusively on this subject. We welcome all skill levels and hope those with greater mathematical knowledge can assist those lacking in it.

If this sounds interesting to you, sing up below and I'll let you know once I've added you. Although the forum may not have much now, that's because I've only just released it, and I hope once I build a community, It will be a very useful and motivating space for those interested in improving their geometry. The link is
here.

I look forward to seeing you all in the forum!
200 replies
fossasor
Jun 10, 2025
Ravensrule8
4 hours ago
JMC/AMC8
hgauri   1
N 5 hours ago by maxd3
wait is the junior maths challenge the same as the amc 8
they both have 25 questions
1 reply
hgauri
5 hours ago
maxd3
5 hours ago
Another Random Problem I Made
Wax_Wax   5
N 5 hours ago by maxd3
Find the shaded area (excluding the right angles):
5 replies
Wax_Wax
Jun 30, 2025
maxd3
5 hours ago
9 Square roots
A7456321   32
N 5 hours ago by sv_legend
Me personally I only have $\sqrt2=1.414$ memorized but I'm sure there are people out there with more!
update: i recently learned $\sqrt3=1.732$

CLICK ON ME YOU KNOW YOU WANT TO
32 replies
A7456321
May 29, 2025
sv_legend
5 hours ago
Cutting square into three congruent rectangles
Silverfalcon   5
N 5 hours ago by sv_legend
A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is

IMAGE

$\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$
5 replies
Silverfalcon
Dec 9, 2005
sv_legend
5 hours ago
half of half of half... of cake
FrancoGiosefAG   3
N 5 hours ago by sv_legend
Victor and Vicky bought a cake and ate it in the following way: One morning Victor ate half of the cake, in the evening Vicky ate the remaining half of the cake. This process continued in the same way for $4$ days, each eating half of the cake they found. On the morning of the fifth day, Victor ate what was left of the cake. What proportion of the cake did Victor eat in the $5$ days?
3 replies
FrancoGiosefAG
Jun 9, 2025
sv_legend
5 hours ago
a,b with ab=n and a+b minimal
FrancoGiosefAG   1
N 5 hours ago by Roots_Of_Moksha
For each positive integer $n$, consider the positive integers $a$ and $b$ such that: $a$ and $b$ have no positive divisors in common other than $1$, $ab = n$, and $a + b$ is minimal (the latter means that if $n = ab = cd$, then $a + b < c + d$.) Let $f(n) = |s(a) - s(b)|$ where $s(j)$ represents the sum of the digits of $j$. Calculate the sum:
\[ f(1^2+1)+f(2^2+2)+\dots+f(2017^2+2017).\]
1 reply
FrancoGiosefAG
Jun 9, 2025
Roots_Of_Moksha
5 hours ago
fractions in fractions
FrancoGiosefAG   3
N 5 hours ago by Roots_Of_Moksha
The positive integers $a,b,c,d,e$ satisfy
\[a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d+\cfrac{1}{e+1}}}}=\frac{268}{187}. \]Find $a+b+c+d+e$.
3 replies
FrancoGiosefAG
Jun 9, 2025
Roots_Of_Moksha
5 hours ago
Find the value of the expression
Darealzolt   5
N 5 hours ago by sv_legend
Let \(x=\frac{1}{2-\sqrt{3}}\), find the value from
\[
x^6-2\sqrt{3}x^5-x^4+x^3-4x^2+2x-\sqrt{3}
\]
5 replies
Darealzolt
6 hours ago
sv_legend
5 hours ago
Zsigmondy's theorem
V0305   24
N Jun 19, 2025 by Mr.Sharkman
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
24 replies
V0305
May 24, 2025
Mr.Sharkman
Jun 19, 2025
Zsigmondy's theorem
G H J
G H BBookmark kLocked kLocked NReply
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V0305
728 posts
#1 • 1 Y
Y by GA34-261
Is Zsigmondy's theorem allowed on the IMO, and is it allowed on the AMC series of proof competitions (e.g. USAJMO, USA TSTST)?
Z K Y
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vincentwant
1473 posts
#2 • 1 Y
Y by GA34-261
yes it is allowed (tstst 2018/8 i think requires it)
Z K Y
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Andyluo
1041 posts
#3 • 1 Y
Y by GA34-261
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?
Z K Y
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CatCatHead
44 posts
#4 • 1 Y
Y by GA34-261
It is allowed and very useful in some special problems
Z K Y
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N3bula
312 posts
#5 • 1 Y
Y by GA34-261
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

Nothing will ever require zsigmondy in a well made contest, note stuff like china contests/rmm are not well made olympiads. The problem you mentioned does not require zsigmondy. Also in terms of citing results if its a named result its citable, be careful with citing stuff from muricaaaa or such tho and it very much depends on what country ur from for what is citable.
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EGMO
158 posts
#6 • 1 Y
Y by GA34-261
buh china contests are decent :(
dissing my home country is wild
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N3bula
312 posts
#7 • 1 Y
Y by GA34-261
EGMO wrote:
buh china contests are decent :(
dissing my home country is wild

China contests are not good or accurate olympiad contests
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ohiorizzler1434
886 posts
#8 • 2 Y
Y by Andyluo, GA34-261
buh we should analyse each contest according to each situation! china contests are unique in their own way and we should not disrespect it for having bad problems!
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pingpongmerrily
3969 posts
#9 • 1 Y
Y by GA34-261
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever
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vincentwant
1473 posts
#10 • 2 Y
Y by aidan0626, GA34-261
pingpongmerrily wrote:
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever

probably because it is very hard to prove
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imagien_bad
74 posts
#11 • 2 Y
Y by bachkieu, GA34-261
xig $         $
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N3bula
312 posts
#12 • 1 Y
Y by GA34-261
pingpongmerrily wrote:
I'm confused, what is special about Zsigmondy's theorem which makes it different from other theorems in terms of being cited? Why can't you just cite it like any other theorem such as the pythagorean theorem or fermat's little theorem or whatever

Harder to prove but not stupid stupid hard either tho
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CatCatHead
44 posts
#13 • 1 Y
Y by GA34-261
Andyluo wrote:
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?

Actually in the TST level you can cite anything if you remember the right place like in lemma 4.13 in the EGMO book written by evan chen balabala. But in the USAJMO/USAMO level you may be marked wrong as there are too much students, so I do not recommond to use not famous lemma in USAJMO and USAMO. (Zsigmondy is famous enough i think)
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N3bula
312 posts
#14 • 1 Y
Y by GA34-261
CatCatHead wrote:
Andyluo wrote:
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

I was curious about citing sources in general.

can you just site anything thats from a book or smth?

Actually in the TST level you can cite anything if you remember the right place like in lemma 4.13 in the EGMO book written by evan chen balabala. But in the USAJMO/USAMO level you may be marked wrong as there are too much students, so I do not recommond to use not famous lemma in USAJMO and USAMO. (Zsigmondy is famous enough i think)

Note that citing aops handouts does not count, if its a published book yes an aops handout so stuff like muricaaaa I dont believe is qoutable.
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Martin2001
170 posts
#15 • 1 Y
Y by GA34-261
Zsig is most definitely allowed, i have used it many times. Remember that the people grading the usajmo/amo and so on are themselves very orz so relatively well known things like Zsig can be used.
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llddmmtt1
431 posts
#17 • 3 Y
Y by aidan0626, GA34-261, MathRook7817
what the zsigma
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dolphinday
1335 posts
#18 • 1 Y
Y by GA34-261
you can use any well known result for olympiad contests. Zsig isn't that obscure
it's also used for 2000 imo p5
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whwlqkd
177 posts
#19 • 1 Y
Y by GA34-261
Everything is allowed if you mention the theorem name and statement.
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Stead
12 posts
#20
Y by
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?
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CatCatHead
44 posts
#21
Y by
Stead wrote:
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?

yeah, actually you can use it in USA TSTST 2024 Q6
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Stead
12 posts
#22
Y by
Oh, nice! Will take a look. Thanks!
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Sudokan
1 post
#23
Y by
What exactly is this theorem? Thank you!
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CatCatHead
44 posts
#24
Y by
Sudokan wrote:
What exactly is this theorem? Thank you!

zsigmondy means that you can always find new primes from a^n-b^n when n change to n+1 every time except fro two specail cases.
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Mr.Sharkman
580 posts
#25
Y by
vincentwant wrote:
yes it is allowed (tstst 2018/8 i think requires it)

China TST 2005/9
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Mr.Sharkman
580 posts
#26
Y by
CatCatHead wrote:
Stead wrote:
I suppose then, if Fermat's Last Theorem would somehow help you make progress in a problem, then it is allowed?

yeah, actually you can use it in USA TSTST 2024 Q6

OH YEAH LOL

That may be the weirded fe ever
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