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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.
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What belongs on this forum?
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Mathcounts and how to learn

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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
Turkish JMO 2025?
bitrak   1
N an hour ago by blug
Let p and q be prime numbers. Prove that if pq(p+ 1)(q + 1)+ 1 is a perfect square, then pq + 1 is also a perfect square.
1 reply
bitrak
Yesterday at 2:04 PM
blug
an hour ago
Combi Algorithm/PHP/..
CatalanThinker   0
an hour ago
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
5. [Czech and Slovak Republics 1997]
Each side and diagonal of a regular n-gon (n ≥ 3) is colored blue or green. A move consists of choosing a vertex and
switching the color of each segment incident to that vertex (from blue to green or vice versa). Prove that regardless of the initial coloring, it is possible to make the number of blue segments incident to each vertex even by following a sequence of moves. Also show that the final configuration obtained is uniquely determined by the initial coloring.
0 replies
CatalanThinker
an hour ago
0 replies
Combi Proof Math Algorithm
CatalanThinker   0
an hour ago
Source: Olympiad_Combinatorics_by_Pranav_A_Sriram
3. [Russia 1961]
Real numbers are written in an $m \times n$ table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
0 replies
CatalanThinker
an hour ago
0 replies
Unexpecredly Quick-Solve Inequality
Primeniyazidayi   1
N an hour ago by Ritwin
Source: German MO 2025,Round 4,Grade 11/12 Day 2 P1
If $a, b, c>0$, prove that $$\frac{a^5}{b^2}+\frac{b}{c}+\frac{c^3}{a^2}>2a$$
1 reply
Primeniyazidayi
an hour ago
Ritwin
an hour ago
Rational equation
soruz   2
N 2 hours ago by Li0nking
Solve the equation:
$ \frac{x+2}{1011} +\frac{x}{1010} +\frac{x-2}{1009} +\frac{x-4}{1008}...+\frac{x-2016}{2}=2020$.
2 replies
soruz
Feb 23, 2022
Li0nking
2 hours ago
Trivial Factoring MathCounts Question? 2025 MathCounts National Sprint Round #29
ilikemath247365   13
N 3 hours ago by Andyluo
What is the value of the expression below?

$\frac{(1! + 2! + 3!)(2! + 3! + 4!)(3! + 4! + 5!)...(98! + 99! + 100!)}{(1! - 3(2!) + 3!)(2! - 3(3!) + 4!)(3! - 3(4!) + 5!)...(98! - 3(99!) + 100!)}$.
13 replies
ilikemath247365
5 hours ago
Andyluo
3 hours ago
9 Favorite topic
A7456321   26
N 5 hours ago by GA34-261
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
26 replies
A7456321
May 23, 2025
GA34-261
5 hours ago
Facts About 2025!
Existing_Human1   263
N Today at 12:10 AM by SpeedCuber7
Hello AOPS,

As we enter the New Year, the most exciting part is figuring out the mathematical connections to the number we have now temporally entered

Here are some facts about 2025:
$$2025 = 45^2 = (20+25)(20+25)$$$$2025 = 1^3 + 2^3 +3^3 + 4^3 +5^3 +6^3 + 7^3 +8^3 +9^3 = (1+2+3+4+5+6+7+8+9)^2 = {10 \choose 2}^2$$
If anyone has any more facts about 2025, enlighted the world with a new appreciation for the year


(I got some of the facts from this video)
263 replies
Existing_Human1
Jan 1, 2025
SpeedCuber7
Today at 12:10 AM
Course completion documents?
Happycat2   3
N Today at 12:00 AM by A7456321
Hi! I just wanted to ask if the course completion documents ( I think that that's what they're called correct me if I'm wrong) does, like what do I need them for?
3 replies
Happycat2
Yesterday at 11:38 PM
A7456321
Today at 12:00 AM
Could I make AIME?
GallopingUnicorn45   74
N Yesterday at 11:41 PM by A7456321
I'm a 4th grader, and I'm about half-way through Intro to Algebra, Intro to C&P, and Intro to Number Theory. I wouldn't say I get all of the material, but I understand like 80-90% of the material. Could I make AIME in 6th or 7th grade? Also, I'm doing AMC 8 for the second time, I got 15 questions last time, would I be able to make Honor or Distinguished Honor Roll this time?
74 replies
GallopingUnicorn45
Dec 11, 2024
A7456321
Yesterday at 11:41 PM
Summer math contest prep
Abby0618   16
N Yesterday at 11:25 PM by Capybara7017
School is almost out, so I have a lot of time in the summer. I want to be able to make DHR on AMC 8 in 7th grade

(current 6th grader) and hopefully get an average score in AMC 10. What should I do during the summer to achieve

these goals? For context, I have many books from AOPS, have already taken the Intro to Algebra A course, and took

AMC 8 for the first time as a 6th grader. If there are any challenging math problems you think would benefit learning,

please post them here. Thank you! :-D
16 replies
Abby0618
May 22, 2025
Capybara7017
Yesterday at 11:25 PM
Vol 1 enough?
Spacepandamath13   26
N Yesterday at 11:16 PM by Capybara7017
is aops vol 1 book enough for amc10 or is vol 2 required to be studied too?
26 replies
Spacepandamath13
May 21, 2025
Capybara7017
Yesterday at 11:16 PM
Solve this
DhruvJha   13
N Yesterday at 11:13 PM by Capybara7017
Len is playing a Arkansas-styled basketball game with his friend, Dawson. The game ends whenever a player has a 2 point lead over the other player. In Arkansas styled basketball, points can only be scored in increments of two. Whenever Len has possession of the ball, he scores at a rate of 60 percent. However, Dawson is slightly worse and when he has possession of the ball, he scores at a rate of only 40 percent. Given that Len starts with possession first each game, what is the expected amount of games he wins if they play 38 total?
13 replies
DhruvJha
May 24, 2025
Capybara7017
Yesterday at 11:13 PM
Combo Bash
DhruvJha   3
N Yesterday at 11:10 PM by Capybara7017
Devin and Cowen are playing a game where they take turns flipping a biased coin. The coin lands on heads with probability 2/3 and tails with probability 1/3. Devin goes first. On each turn, the current player flips the coin repeatedly until the coin lands tails. For each heads flipped, the player gains 1 point and continues flipping. If the coin lands tails, their turn ends, and the other player takes their turn. The first player to reach 3 points wins the game immediately. What is the probability that Devin wins the game? Express your answer as a common fraction in lowest terms.
3 replies
DhruvJha
Yesterday at 3:18 AM
Capybara7017
Yesterday at 11:10 PM
Easy Combinatorial Game Problem in Taiwan TST
chengbilly   8
N Apr 30, 2025 by CrazyInMath
Source: 2025 Taiwan TST Round 1 Independent Study 1-C
Alice and Bob are playing game on an $n \times n$ grid. Alice goes first, and they take turns drawing a black point from the coordinate set
\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers $n$ such that Alice has a winning strategy.

Proposed by chengbilly
8 replies
chengbilly
Mar 5, 2025
CrazyInMath
Apr 30, 2025
Easy Combinatorial Game Problem in Taiwan TST
G H J
G H BBookmark kLocked kLocked NReply
Source: 2025 Taiwan TST Round 1 Independent Study 1-C
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chengbilly
11 posts
#1 • 1 Y
Y by Rounak_iitr
Alice and Bob are playing game on an $n \times n$ grid. Alice goes first, and they take turns drawing a black point from the coordinate set
\[\{(i, j) \mid i, j \in \mathbb{N}, 1 \leq i, j \leq n\}\]There is a constraint that the distance between any two black points cannot be an integer. The player who cannot draw a black point loses. Find all integers $n$ such that Alice has a winning strategy.

Proposed by chengbilly
This post has been edited 1 time. Last edited by chengbilly, Mar 5, 2025, 5:09 AM
Z K Y
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CHESSR1DER
69 posts
#2
Y by
1) $n = 2k+1$
Let's put center of coordinates in $(k+1;k+1)$.
Alice will mark $(0,0)$. Then if Bob marks $(a,b)$ then
$a^2+b^2$ is not a square. Alice can mark $(-a,-b)$ since points are symmetrical with respect to $(0,0)$, then we only need to check if distance from $(-a,-b)$ to $(a,b)$ is an integer.
But $d^2 = (2a)^2 + (2b)^2 = 4(a^2+b^2) $is not a square. So if $n = 2k+1$ then Alice wins.
2) $n=2k$
Lets put the center of coordinates in $(k+0,5;k+0,5)$.

If Alice marks $(a+0,5;b+0,5)$ then Bob should mark $(-a-0,5;-b-0,5)$. Points are symmetrical with respect to the point $(0,0)$. So we only need to check if $d^2$ is not a square.
But $d^2 = (2a+1)^2 + (2b+1)^2 \equiv 2 (mod 4)$. So $d^2$ is not a square. So if $n = 2k$ then Bob wins.
This post has been edited 1 time. Last edited by CHESSR1DER, Mar 5, 2025, 7:21 PM
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alba_tross1867
44 posts
#3 • 1 Y
Y by Hamzaachak
Yet another symmetry problem.
For $n$ odd, Alice plays center of the grid, and then copy (do symmetric of ) Bob's moves.
For $n$ even, Bob plays symmetric of Alice's moves.
Z K Y
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nguyenhuybao_06
39 posts
#4
Y by
Actually, if $n=2k+1,$ Alice can play any point, but follow the rule; then she'll win. And if $n=2k,$ Bob can play any point, but follow the rule; then he'll win. Cuz the problem is equivalent to: pick any point and draw all point on its row and column by black. The game must end after $n$ turns.
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CHESSR1DER
69 posts
#5
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nguyenhuybao_06 wrote:
Cuz the problem is equivalent to: pick any point and draw all point on its row and column by black. The game must end after $n$ turns.

Its not equal to this as for example points $(0,0)$ and $(3,4)$ both can't be marked.
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GreekIdiot
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#6
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$\cdots$
This post has been edited 2 times. Last edited by GreekIdiot, May 7, 2025, 4:08 PM
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CHESSR1DER
69 posts
#7
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Quote:
Consider $(a,b)=(3,4)$ then notice $a^2+b^2=25$ is a perfect square
Same goes for all pythagorean triples
If $a^2+b^2$ is a square, then Bob can't mark it as (0,0) already marked.
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cursed_tangent1434
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#8
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Extremely standard idea, but the problem is quite enjoyable nevertheless. We claim that Alice wins if and only if $n$ is odd and Bob wins if and only if $n$ is even. We tackle each of these cases separately. Let a violating pair denote a pair of points which have an integer distance between them.

Case 1 : $n$ is odd. In this case, let the center square $O$ of the grid denote cell $\left(\frac{n+1}{2},\frac{n+1}{2}\right)$. Alice's strategy is simple. In here first move she draws a black point at the center square, and then reflects Bob's move across the center square.

To see why this works, assume that Alice cannot copy Bob's move after a certain point. If Bob draws a black point at $P$, then $P'$ must be either already colored black or an integer distance away from a black point. The former is impossible due to the symmetric nature of the moves until this point. For the latter to be possible the violating pair must be either $(P',O)$ , $(P,P')$ or $(P',Q)$ where $Q \not \in \{P,P',O\}$. Clearly if $(P',O)$ is a violating pair by symmetry so is $(P,O)$ implying that Bob performed an illegal move in his previous move. Further if $(P,P')$ is a violating pair so must be $(P,O)$ since an easy calculation yields that $PP'=2PO$. Further, if $(P',Q)$ were a violating pair, then $(P,Q')$ must be a violating pair where $Q'$ is the reflection of $Q$ across the center square. But by the nature of the algorithm, $Q'$ must already be marked which implies that Bob's previous move was illegal.

Thus, as long as Bob has a legal move, so does Alice and since the number of cells on the board is odd, it is Bob who runs out of legal moves first.

Case 2 : $n$ is even. This case is almost entirely similar however the center point $O$ is now defined as the center of the square formed by points $\left(\frac{n}{2},\frac{n}{2}\right)$ , $\left(\frac{n}{2},\frac{n}{2}+1\right)$, $\left(\frac{n}{2}+1,\frac{n}{2}\right)$ and $\left(\frac{n}{2}+1,\frac{n}{2}+1\right)$. Bob now simply performs the symmetric move to Alice's across the center point in each move. If he is unable to do this at any point, say after Alice draws the black point $P$ then either $(P,P')$ or $(P,'Q)$ for $Q\ne P$ must be a violating pair. In the later case, the reflection of $Q$ across the center point $Q'$ must already have been marked due to the nature of the given algorithm implying that $(P,Q')$ is already a violating pair invalidating Alice's final move.

In the former case, note that the horizontal and vertical distances between points $P$ and $P'$ are odd by symmetry. Applying the Pythagorean Theorem results in $PP' = \sqrt{a^2+b^2}$ for odd positive integers $a$ and $b$ which implies that the quantity under the square root is $2\pmod{4}$ which can never be a perfect square. Thus, this distance is not an integer and hence it cannot be a violating pair. Thus, as long as Alice has a legal move, so does Bob and since the number of cells on the board is even, it is Alice who runs out of legal moves first.
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CrazyInMath
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#9
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solution
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