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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
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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
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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
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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
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What belongs on this forum?
How do I write a thorough solution?
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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
J
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Proving radical axis through orthocenter
azzam2912   0
12 minutes ago
In acute triangle $ABC$ let $D, E$ and $F$ denote the feet of the altitudes from $A, B$ and $C$, respectively. Let line $DE$ intersect circumcircle $ABC$ at points $G, H$. Similarly, let line $DF$ intersect circumcircle $ABC$ at points $I, J$. Prove that the radical axis of circles $EIJ$ and $FGH$ passes through the orthocenter of triangle $ABC$
0 replies
azzam2912
12 minutes ago
0 replies
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Ez induction to start it off
alexanderhamilton124   22
N 18 minutes ago by Adywastaken
Source: Inmo 2025 p1
Consider the sequence defined by \(a_1 = 2\), \(a_2 = 3\), and
\[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \]for all integers \(k \geq 1\). Determine all positive integers \(n\) such that
\[ \frac{a_n}{n} \]is an integer.

Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.
22 replies
alexanderhamilton124
Jan 19, 2025
Adywastaken
18 minutes ago
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Weird Algebra?
JARP091   0
19 minutes ago
Source: Art and Craft of Problem Solving 2.2.16
For each positive integer \( n \), find positive integer solutions \( x_1, x_2, \ldots, x_n \) to the equation

\[ \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} + \frac{1}{x_1 x_2 \cdots x_n} = 1 \]
0 replies
JARP091
19 minutes ago
0 replies
J
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Parallel lines in incircle configuration
GeorgeRP   2
N 23 minutes ago by bin_sherlo
Source: Bulgaria IMO TST 2025 P1
Let $I$ be the incenter of triangle $\triangle ABC$. Let $H_A$, $H_B$, and $H_C$ be the orthocenters of triangles $\triangle BCI$, $\triangle ACI$, and $\triangle ABI$, respectively. Prove that the lines through $H_A$, $H_B$, and $H_C$, parallel to $AI$, $BI$, and $CI$, respectively, are concurrent.
2 replies
GeorgeRP
4 hours ago
bin_sherlo
23 minutes ago
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MathCounts National Sets
UberPiggy   14
N Today at 5:15 AM by MathRook7817
Hi, does anyone happen to have MathCounts National round test problems from 2018-2024? I found 2000-2017 online but can't find anything past that year.
14 replies
UberPiggy
Mar 28, 2025
MathRook7817
Today at 5:15 AM
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MOPAMCAIMEUSAMOAMC
JustKeepRunning   9
N Today at 4:28 AM by Craftybutterfly
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text{USAMO}$. He will need to get at least $200$ points on $\text{the AMC}$ and $\text{AIME}$ combined and get at least $21$ points on $\text{the USAMO}$ to make $\text{MOP}$. What is the minimum amount of effort he can expend to make $\text{MOP}$?
9 replies
JustKeepRunning
Jul 27, 2019
Craftybutterfly
Today at 4:28 AM
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2025 Mathcounts Nationals Journal
Andyluo   12
N Today at 4:24 AM by jkim0656
Friday May 9th

I spent my evening after school, packing for the trip, using the checklist given by my coach.
I didn’t do much preparation, as I was mostly chilling out for the upcoming days.

I also played basketball with my cousin, Kevin, who met Gotham Chess and stayed at his home!


Saturday, May 10th

I woke up at 5:30 AM, ate a light breakfast, and headed out to the airport with my luggage.

I met my teacher, but was surprised that Archishman split up with his own family.
Waiting for the TSA was pretty boring, but we soon got through, and after I found our gate.

A couple of minutes pass by, as I review an AOPS mock where I meet Archischman;
Afterward, we chill out, watch the rube goldberg machine in the airport, and wait to board the plane.

During the plane ride, I played games; however, during our descent, I heard a loud crack, and our plane started wobbling, and we heard cracking sounds in the seats. Fortunately, we were able to land and were able to attend the competition the next day.

After this, heading out, we went to the shuttle; however, we had 35 minutes. We tried to solve the Jane Street card puzzle but failed, and ended up socializing.

After we arrived at the hotel, we received a MASSIVE amount of stuff, like calculators, shirts, coupons, plaques, stickers, etc.
I also saw and got a signature from Richard Rusczyk, which was really cool.

Then, we went to a restaurant named “Chinatown Garden”, with the worst food I’ve ever had.

We then chilled in our rooms, studied for a bit, and started organizing plans for pin trading.

Our goal was to scam as many people as possible by doing 2:1 trades, as we had a “limited”
amount of pins. (We even got 5:1 and 10:1 trades)
A Virginia kid scammed me with a STEM pin, so I chased him down and got our pin back.

We got through around half the states organizing in and out of what pins we had.

Finally, we got some food from the buffet (which was surprisingly decent) and had a good time trading some more.

We ended the day with a short and brief CDR, where we had some fun, and then we went to sleep to anticipate the next day.

At night, I showered and sang karaoke with Archi.

Sunday, May 11th

Getting ready, I found out that a mock (outside the box) was recently released and took it through breakfast.

Then, once we got there at 8:30, there was a mob of parents taking pictures, and music played.

Then every team did introductions/attendance and their chants, most of which were really cringe.

I took the test; however was too slow on the sprint round and got a predicted 16.

On the target round, I was able to get through and got a 12, despite barely not solving p8 to my frustration.

Team round we did decently, scoring a 14/20, which was one of the best scores around us, that even orz states like Texas and Washington didn’t beat.

Predicted Scores:
Caleb and I got a similar score (around 28), and Henry got around 35, and Archi sold on the target and got a 24.

After this, we teamed up with North Carolina (chill af) and went to a pho shop (54 Restaurant), which tasted amazing. (A far contrast from Chinatown Garden)

Then, we went to an aerospace museum, where we played Brawl Stars and went around. Eventually, we saw models of blackholes and air vacuums, and played a flight simulator.

Then we went to our hotel, chilled, and watched basketball games.

After, we went to an Indian restaurant named “Himilayan Doko” which was really delicious!

Then we raided different rooms, from NC, HAWAII, Idaho, Virgin Islands, and accidentally a random dudes room who was ticked at us.

Finally, we chilled and went to sleep, though I tried to get Henry and Archi to sleep since they were being annoying.

Monday, May 12th

We start the day forming my pin badge, and then we went to get some breakfast.

After that, we met in the breakfast area with 2 teams for table, and I actually got a 10:1 pin trade which was pretty cool.

After that, we lined up and got our thunderstick/clapping machines, and ran through the entrance of the CDR.

Sadly, we didn’t win anything, but it was cool seeing the results.

Then, we started to watch the CDR, which was really exciting.
It got really interesting when everyone saw Nathan Liu cook his opponent in half a second.

In the semifinals, it was insane, and Advait and Nathan, buzzed every question that was around mid-sprint level.

Then, it finished with Nathan beating Brandon with a 2-second solve, absolute insanity.

Finally, we went back to our rooms and got lunch in the hotel.
A few hours later, we received our scores, and I had bombed, scoring a 26 with 7 sillies. (ouch)

Unfortunately, my teammates Henry and Archishman sillied a bunch of questions.

After, we played Brawl Stars, and went to explore the hotel, where we went up a random staircase and got stuck. We went to the roof, but got scared and yelled out for help on the gym floor. Thankfully, we got back, and I went and reviewed the test.

After we reviewed the test, and went to the Mathcounts Party.

The food was mid, but the games were pretty fun.

We met a bunch of people, played air hockey, foosball, and basketball, while listening to the not so great music in the background.

Then we went back to our rooms at 8PM, to put our pins on, and I got 38/56!

Finally, we met up in a room with a bunch of Cali, and NC kids, and talked about the test, the people, and played Brawl Stars. Even Josh Frost came up to us and asked us how the trip was.

Tuesday, May 13th

I started the day waking up at 6:20, and packed up and ate breakfast. After that, Henry was late, so we packed food for him and went to the bus shuttle.

Eventually, we arrived at the airport, went through security (which was suspiciously fast), and played Brawl Stars. We also ate five guys fries, which was pretty good. Eventually, we had to part our ways with Henry and headed out to our flights, which marked the end of the trip.

Conclusion:

Although we didn’t do amazingly well in the contest, going to DC was an amazing experience. I got to meet people who were passionate about math, and hang out with them, goofing around.

This was the best math contest experience that I’ll likely ever have, and I’m glad I went through it.
12 replies
Andyluo
Yesterday at 4:43 PM
jkim0656
Today at 4:24 AM
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9 MathandAI4Girls!!!
Inaaya   26
N Today at 2:22 AM by fossasor
How many problems did y'all solve this year?
I clowned and started the pset the week before :oops:
Though I think if i used the time wisely, I could have at least solved 11 of them
ended up with 9 :wallbash_red:
26 replies
Inaaya
May 7, 2025
fossasor
Today at 2:22 AM
J
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The daily problem!
Leeoz   161
N Today at 1:38 AM by Math-lover1
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!
161 replies
Leeoz
Mar 21, 2025
Math-lover1
Today at 1:38 AM
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MAP Goals
Antoinette14   14
N Today at 1:27 AM by Schintalpati
What's yall's MAP goals for this spring?
Mine's a 300 (trying to beat my brother's record) but since I'm at a 285 rn, 290+ is more reasonable.
14 replies
Antoinette14
May 8, 2025
Schintalpati
Today at 1:27 AM
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Berkeley mini Math Tournament Online is June 7
BerkeleyMathTournament   8
N Today at 12:55 AM by jb2015007
Berkeley mini Math Tournament is a math competition hosted for middle school students once a year. Students compete in multiple rounds: individual round, team round, puzzle round, and relay round.

BmMT 2025 Online will be held on June 7th, and registration is OPEN! Registration is $8 per student. Our website https://berkeley.mt/events/bmmt-2025-online/ has more details about the event, past tests to practice with, and frequently asked questions. We look forward to building community and inspiring students as they explore the world of math!

3 out of 4 of the rounds are completed with a team, so it’s a great opportunity for students to work together. Beyond getting more comfortable with math and becoming better problem solvers, our team is preparing some fun post-competition activities!

Registration is open to students in grades 8 or below. You do not have to be local to the Bay Area or California to register for BmMT Online. Students may register as a team of 1, but it is beneficial to compete on a team of at least 3 due to our scoring guideline and for the experience.

We hope you consider attending, or if you are a parent or teacher, that you encourage your students to think about attending BmMT. Thank you, and once again find more details/register at our website.
8 replies
BerkeleyMathTournament
May 1, 2025
jb2015007
Today at 12:55 AM
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Mathcounts Nationals Written Score Hub
DhruvJha   52
N Today at 12:55 AM by c_double_sharp
Put in your estimated score on the written nats comp on Sunday after the comp so we can get a good idea of the cdr quals are
52 replies
DhruvJha
May 10, 2025
c_double_sharp
Today at 12:55 AM
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problemo
hashbrown2009   5
N Today at 12:49 AM by ZMB038
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.
5 replies
hashbrown2009
Mar 30, 2025
ZMB038
Today at 12:49 AM
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Camp Conway acceptance
fossasor   22
N Today at 12:48 AM by ZMB038
Hello! I've just been accepted into Camp Conway, but I'm not sure how popular this camp actually is, given that it's new. Has anyone else applied/has been accepted/is going? (I'm trying to figure out to what degree this acceptance was just lack of qualified applicants, so I can better predict my chances of getting into my preferred math camp.)
22 replies
fossasor
Feb 20, 2025
ZMB038
Today at 12:48 AM
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Inspired by old results
sqing   6
N Apr 29, 2025 by sqing
Source: Own
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
6 replies
sqing
Apr 26, 2025
sqing
Apr 29, 2025
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sqing
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sqing
#1 h  Apr 26, 2025, 12:30 PM
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Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{2}{abc}\geq 2+\sqrt 3$$$$ \frac{3}{a}+\frac {3}{ab}+\frac{1}{abc}\geq\frac {7+\sqrt {13}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{3}{abc}\geq\frac {5+\sqrt {21}}{2}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{4}{abc}\geq 3+2\sqrt 2$$
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sqing
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sqing
#2 h  Apr 26, 2025, 12:42 PM
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Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{1}{a}+\frac {1}{ab}+\frac{1}{ca}+\frac{1}{abc}\geq \frac {33+7\sqrt {21}}{18}$$$$ \frac{1}{a}+\frac {1}{ab}+\frac{1}{bc}+\frac{1}{ca}+\frac{1}{abc}\geq \frac {5+4\sqrt[3] {2}+3\sqrt[3] {4}}{3}$$
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ErTeeEs06
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ErTeeEs06
#3 h  Apr 26, 2025, 8:30 PM
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sqing wrote:
Let $ a,b,c>0 $ and $ a+b+c=3. $ Prove that
$$ \frac{2}{a}+\frac {2}{ab}+\frac{1}{abc}\geq 4$$
I solved it a bit ugly by substituting $x=2a-1, y=2b, z=2c$. The inequality then becomes prove $z+1\leq \frac{xyz}{2}$ for positive real numbers $x, y, z$ with sum 5. (We can assume $x>0$ since for $a<\frac{1}{2}$ the inequality is trivial). Now here we can WLOG take $x=y$ and subsitute $z=5-2x$. Then the inequality reduces to $(x-2)^2(2x+3)\geq 0$ which is obvious. Does anyone have a nicer solution?
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#4 h  Apr 27, 2025, 1:51 AM
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Good.Thanks.
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Sunjee
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#5 h  Apr 29, 2025, 11:30 AM
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I will solve only
$$ \frac{1}{a}+\frac{1}{ab}+\frac{3}{abc}\geq \frac{5+\sqrt{21}}{2} $$Rest is similar. Using AM-GM Inequality, we have
\begin{align*} \frac{1}{a}+\frac{a}{x^2}&\geq \frac{2}{x},\\ \frac{1}{ab}+\frac{a}{x^2y}+\frac{b}{xy^2}&\geq \frac{3}{xy},\\ \frac{3}{abc}+\frac{3a}{x^2yz}+\frac{3b}{xy^2z}+\frac{3c}{xyz^2}&\geq \frac{12}{xyz} \end{align*}Choosing here, $x=\dfrac{\sqrt{7}(\sqrt{7}-\sqrt{3})}{2}$, $y=1,$ $z=\dfrac{\sqrt{3}(\sqrt{7}-\sqrt{3})}{2}$ and summing we get result.
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Sunjee
529 posts
Sunjee
#6 h  Apr 29, 2025, 11:44 AM
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First one is easiest one. Using AM-GM Inequality we have
\begin{align*} \frac{2}{a}+\frac{8}{9}a&\geq \frac{8}{3},\\ \frac{2}{ab}+\frac{8}{9}a+\frac{4}{3}b&\geq 4,\\ \frac{1}{abc}+\frac{8}{9}a+\frac{4}{3}b+\frac{8}{3}c&\geq \frac{16}{3} \end{align*}Adding these inequalities we get
$$\frac{2}{a}+\frac{2}{ab}+\frac{1}{abc}\geq 4$$
This post has been edited 1 time. Last edited by Sunjee, Apr 29, 2025, 11:44 AM
Reason: mis type
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sqing
#7 h  Apr 29, 2025, 12:18 PM
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Very nice.Thanks.
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