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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
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?
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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
Easy P4 combi game with nt flavour
Maths_VC   1
N 39 minutes ago by p.lazarov06
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
1 reply
Maths_VC
May 27, 2025
p.lazarov06
39 minutes ago
Central sequences
EeEeRUT   14
N an hour ago by HamstPan38825
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
14 replies
EeEeRUT
Apr 16, 2025
HamstPan38825
an hour ago
Elementary Problems Compilation
Saucepan_man02   32
N 2 hours ago by atdaotlohbh
Could anyone send some elementary problems, which have tricky and short elegant methods to solve?

For example like this one:
Solve over reals: $$a^2 + b^2 + c^2 + d^2  -ab-bc-cd-d +2/5=0$$
32 replies
Saucepan_man02
May 26, 2025
atdaotlohbh
2 hours ago
Random Points = Problem
kingu   5
N 2 hours ago by happypi31415
Source: Chinese Geometry Handout
Let $ABC$ be a triangle. Let $\omega$ be a circle passing through $B$ intersecting $AB$ at $D$ and $BC$ at $F$. Let $G$ be the intersection of $AF$ and $\omega$. Further, let $M$ and $N$ be the intersections of $FD$ and $DG$ with the tangent to $(ABC)$ at $A$. Now, let $L$ be the second intersection of $MC$ and $(ABC)$. Then, prove that $M$ , $L$ , $D$ , $E$ and $N$ are concyclic.
5 replies
kingu
Apr 27, 2024
happypi31415
2 hours ago
SOLVE: CDR style problem quick algebra
ryfighter   3
N 5 hours ago by cheltstudent
It takes 3 people 10 minutes to mow 2 lawns. How many minutes will it take for 2 people to mow 10 lawns? Express your answer in hours as a decimal.

$(A)$ $1.25$
$(B)$ $75$
$(C)$ $01.025$
$(D)$ $1.5$
$(E)$ $15.25$
3 replies
ryfighter
Today at 3:19 AM
cheltstudent
5 hours ago
Math with Connect4 Boards
Math-lover1   12
N 5 hours ago by Math-lover1
Hi! So I was playing Connect4 with my friends the other day and I wondered: how many "legal" arrangements of Connect4 can be reached at the ending position?

We assume that we do not stop the game when there is a four in a row, and we have 21 red pieces and 21 yellow pieces. We also drop the pieces one by one into a standard 7 by 6 board. We can start the game with any color piece.

https://en.wikipedia.org/wiki/Connect_Four

Initial Thoughts
Attempt to use one-to-one correspondences
12 replies
Math-lover1
May 1, 2025
Math-lover1
5 hours ago
Challenge: Make every number to 100 using 4 fours
CJB19   272
N 5 hours ago by bbojy
I've seen this attempted a lot but I want to see if the AoPS community can actually do it. Using ONLY 4 fours and math operations, make as many numbers as you can. Try to go in order. I'll start:
$$(4-4)*4*4=0$$$$4-4+4/4=1$$$$4/4+4/4=2$$$$(4+4+4)/4=3$$$$4+(4-4)*4=4$$$$4+4^{4-4}=5$$$$4!/4+4-4=6$$$$4+4-4/4=7$$$$4+4+4-4=8$$
272 replies
CJB19
May 15, 2025
bbojy
5 hours ago
The daily problem!
Leeoz   216
N Today at 1:42 PM by kjhgyuio
Every day, I will try to post a new problem for you all to solve! If you want to post a daily problem, you can! :)

Please hide solutions and answers, hints are fine though! :)

Problems usually get harder throughout the week, so Sunday is the easiest and Saturday is the hardest!

Past Problems!
216 replies
Leeoz
Mar 21, 2025
kjhgyuio
Today at 1:42 PM
Geometry question !
kjhgyuio   1
N Today at 11:13 AM by whwlqkd
........
1 reply
kjhgyuio
Today at 10:13 AM
whwlqkd
Today at 11:13 AM
Overly wordy problems
ZMB038   11
N Today at 5:48 AM by Yiyj
Hey everyone, here we can post questions with way to many extraneous words, that are actually easy.
Try to solve the one above yours.
I'll start:
Click to reveal hidden text
11 replies
ZMB038
May 28, 2025
Yiyj
Today at 5:48 AM
Find the amount of possible values from the expression
Darealzolt   3
N Today at 3:16 AM by Justbrick
Find the amount of possible values from
\[
\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}
\]For all non zero integers \(a,b,c\)
3 replies
Darealzolt
Yesterday at 4:15 AM
Justbrick
Today at 3:16 AM
Area of Polygon
AIME15   54
N Today at 2:44 AM by CJB19
The area of polygon $ ABCDEF$, in square units, is

IMAGE

\[ \textbf{(A)}\ 24 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 46 \qquad
\textbf{(D)}\ 66 \qquad
\textbf{(E)}\ 74
\]
54 replies
AIME15
Jan 12, 2009
CJB19
Today at 2:44 AM
2500th Post!
PikaPika999   21
N Today at 2:13 AM by PikaPika999
I may be a bit late for this, but this is my 2500th post :)
Also this is going to be my last one until another big milestone bc I don't wanna clog up the MSM forum with my milestones

Also, since my 1000th post math story was locked due to a flamewar, here is my math story with a few updates :)
(This was also scripted so if there are any problems in my story, um... well, it is what it is)

Script starts:


When I had less than 25 posts on AoPS, I saw many people create threads about them getting 1000th posts and their math story. I thought I would never hit 1000 posts, but I did, and that thread got locked...Please


So, lol here we are, writing my math story again :)


Daycare

Preschool

Kindergarten

First Grade

Second Grade

Third Grade

Fourth Grade

Fifth Grade

Sixth Grade

In conclusion, AoPS has helped me improve my math. Minor side note, but

Finally, I would like to say thank you to all the new friends I made and all the instructors on AoPS that taught me!

Another minor side note, but

and here are some problems ig :)

Problems

hopefully these problems weren't too easy lol
21 replies
PikaPika999
Thursday at 10:25 PM
PikaPika999
Today at 2:13 AM
9 Favorite topic
A7456321   33
N Today at 1:45 AM by A7456321
What is your favorite math topic/subject?

If you don't know why you are here, go binge watch something!

If you forgot why you are here, go to a hospital! :)

If you know why you are here and have voted, maybe say why you picked the option that you picked in a response) :thumbup:

if ur here for any reason whatsoever, CLICK ME YOU KNOW YOU WANT TO
Timeline
33 replies
A7456321
May 23, 2025
A7456321
Today at 1:45 AM
circumcircle of KHM, tangent to altitude, orthocenter and midpoints related
parmenides51   6
N Jun 25, 2020 by khina
Source: Russian Olympiad 2017 HSO 11.6 High Standards Olympiad - Высшая проба
The heights $AA_1, BB_1, CC_1$ of the acute-angled triangle $ABC$ intersect at point $H$. Let $M$ be the midpoint of $BC, K$ the midpoint of $B_1C_1$. Prove that the circle passing through $K, H$ and $M$ is tangent to $AA_1$.
IMAGE
6 replies
parmenides51
Jun 8, 2020
khina
Jun 25, 2020
circumcircle of KHM, tangent to altitude, orthocenter and midpoints related
G H J
Source: Russian Olympiad 2017 HSO 11.6 High Standards Olympiad - Высшая проба
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parmenides51
30653 posts
#1
Y by
The heights $AA_1, BB_1, CC_1$ of the acute-angled triangle $ABC$ intersect at point $H$. Let $M$ be the midpoint of $BC, K$ the midpoint of $B_1C_1$. Prove that the circle passing through $K, H$ and $M$ is tangent to $AA_1$.
https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi9iLzNjOTAzMWYwMzJiYjgzY2YxYzViYmQ4ZjYyNDYyMTAzYzc3MTg3LmdpZg==&rn=dmlzaHByb2JhMjAxNy0xMS02YS5naWY=
This post has been edited 2 times. Last edited by parmenides51, Jun 8, 2020, 8:50 PM
Z K Y
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Steve12345
620 posts
#2 • 1 Y
Y by Mango247
15 minute bash
Here's a nice solution:
Note that the problem is equivalent to this (taking the $AB1C1$ as the starting triangle):
Given triangle $ABC$ and circumcenter $O$ and A-Antipode $A'$. Let $S$ be the intersection of the tangets at $B$ and $C$ and $M$ be the midpoint of $BC$. Note that $OM=R \cdot cos\alpha$ and $OS=\frac{R}{cos\alpha}$ thus $OA'^2=R^2=OS*OM$ so basically we could pick any point not just A-Antipode.
This post has been edited 2 times. Last edited by Steve12345, Jun 8, 2020, 9:04 AM
Z K Y
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Ricochet
144 posts
#3 • 3 Y
Y by Mango247, Mango247, Mango247
First let's prove something, let $N$ be the midpoint of $AH$, then $M, K, N$ are collinear, indeed let $K'=MN\cap B_1C_1$ and by since $M, N$ are the centers of $(BCB_1C_1), (AB_1HC_1)$ resp. by radical axis we have that $MN$ is the perpendicular bisector of $B_1C_1$ and $K\equiv K'$ now let $D=AM\cap B_1C_1$ and $E=AK\cap BC$, notice that $AM, AK$ are isogonal about $\angle BAC$, then $\angle ADK=\angle DAB_1+\angle DB_1A=\angle EAB+\angle EBA=\angle KEM$ and $KDME$ is cyclic and $\angle DEM=90^{\circ}$. With this we see that $\angle A_1AM=\angle EDM=\angle EKM =\angle NKA$ and $\angle NAK=\angle KED=\angle KMD$ then $NA^2=NK\cdot NM=NH^2$ and we are done.
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parmenides51
30653 posts
#4
Y by
a figure by zuss77 from here
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jayme
9801 posts
#5
Y by
Dear Mathlinkers,

http://jl.ayme.pagesperso-orange.fr/Docs/Orthique%20encyclopedie%206.pdf p. 38...

Sincerely
Jean-Louis
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MarkBcc168
1595 posts
#6 • 1 Y
Y by amar_04
15 seconds solution :D :D

Let $N$ be the midpoint of $AH$, i.e. the center of $\omega = \odot(AB_1C_1H)$. By a well-known fact, $MB_1$ and $MC_1$ are tangent to $\omega$. Thus $\overline{NMK}$ is the perpendicular bisector of $B_1C_1$ and $NK\cdot KM = NB_1^2 = NH^2$, done.
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khina
995 posts
#7 • 4 Y
Y by kiyoras_2001, Mango247, Mango247, Mango247
wait isn't this just a specific version of isl 2009 g4?
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