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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Peer-to-Peer Programs Forum
jwelsh   157
N Dec 11, 2023 by cw357
Many of our AoPS Community members share their knowledge with their peers in a variety of ways, ranging from creating mock contests to creating real contests to writing handouts to hosting sessions as part of our partnership with schoolhouse.world.

To facilitate students in these efforts, we have created a new Peer-to-Peer Programs forum. With the creation of this forum, we are starting a new process for those of you who want to advertise your efforts. These advertisements and ensuing discussions have been cluttering up some of the forums that were meant for other purposes, so we’re gathering these topics in one place. This also allows students to find new peer-to-peer learning opportunities without having to poke around all the other forums.

To announce your program, or to invite others to work with you on it, here’s what to do:

1) Post a new topic in the Peer-to-Peer Programs forum. This will be the discussion thread for your program.

2) Post a single brief post in this thread that links the discussion thread of your program in the Peer-to-Peer Programs forum.

Please note that we’ll move or delete any future advertisement posts that are outside the Peer-to-Peer Programs forum, as well as any posts in this topic that are not brief announcements of new opportunities. In particular, this topic should not be used to discuss specific programs; those discussions should occur in topics in the Peer-to-Peer Programs forum.

Your post in this thread should have what you're sharing (class, session, tutoring, handout, math or coding game/other program) and a link to the thread in the Peer-to-Peer Programs forum, which should have more information (like where to find what you're sharing).
157 replies
jwelsh
Mar 15, 2021
cw357
Dec 11, 2023
k i C&P posting recs by mods
v_Enhance   0
Jun 12, 2020
The purpose of this post is to lay out a few suggestions about what kind of posts work well for the C&P forum. Except in a few cases these are mostly meant to be "suggestions based on historical trends" rather than firm hard rules; we may eventually replace this with an actual list of firm rules but that requires admin approval :) That said, if you post something in the "discouraged" category, you should not be totally surprised if it gets locked; they are discouraged exactly because past experience shows they tend to go badly.
-----------------------------
1. Program discussion: Allowed
If you have questions about specific camps or programs (e.g. which classes are good at X camp?), these questions fit well here. Many camps/programs have specific sub-forums too but we understand a lot of them are not active.
-----------------------------
2. Results discussion: Allowed
You can make threads about e.g. how you did on contests (including AMC), though on AMC day when there is a lot of discussion. Moderators and administrators may do a lot of thread-merging / forum-wrangling to keep things in one place.
-----------------------------
3. Reposting solutions or questions to past AMC/AIME/USAMO problems: Allowed
This forum contains a post for nearly every problem from AMC8, AMC10, AMC12, AIME, USAJMO, USAMO (and these links give you an index of all these posts). It is always permitted to post a full solution to any problem in its own thread (linked above), regardless of how old the problem is, and even if this solution is similar to one that has already been posted. We encourage this type of posting because it is helpful for the user to explain their solution in full to an audience, and for future users who want to see multiple approaches to a problem or even just the frequency distribution of common approaches. We do ask for some explanation; if you just post "the answer is (B); ez" then you are not adding anything useful.

You are also encouraged to post questions about a specific problem in the specific thread for that problem, or about previous user's solutions. It's almost always better to use the existing thread than to start a new one, to keep all the discussion in one place easily searchable for future visitors.
-----------------------------
4. Advice posts: Allowed, but read below first
You can use this forum to ask for advice about how to prepare for math competitions in general. But you should be aware that this question has been asked many many times. Before making a post, you are encouraged to look at the following:
[list]
[*] Stop looking for the right training: A generic post about advice that keeps getting stickied :)
[*] There is an enormous list of links on the Wiki of books / problems / etc for all levels.
[/list]
When you do post, we really encourage you to be as specific as possible in your question. Tell us about your background, what you've tried already, etc.

Actually, the absolute best way to get a helpful response is to take a few examples of problems that you tried to solve but couldn't, and explain what you tried on them / why you couldn't solve them. Here is a great example of a specific question.
-----------------------------
5. Publicity: use P2P forum instead
See https://artofproblemsolving.com/community/c5h2489297_peertopeer_programs_forum.
Some exceptions have been allowed in the past, but these require approval from administrators. (I am not totally sure what the criteria is. I am not an administrator.)
-----------------------------
6. Mock contests: use Mock Contests forum instead
Mock contests should be posted in the dedicated forum instead:
https://artofproblemsolving.com/community/c594864_aops_mock_contests
-----------------------------
7. AMC procedural questions: suggest to contact the AMC HQ instead
If you have a question like "how do I submit a change of venue form for the AIME" or "why is my name not on the qualifiers list even though I have a 300 index", you would be better off calling or emailing the AMC program to ask, they are the ones who can help you :)
-----------------------------
8. Discussion of random math problems: suggest to use MSM/HSM/HSO instead
If you are discussing a specific math problem that isn't from the AMC/AIME/USAMO, it's better to post these in Middle School Math, High School Math, High School Olympiads instead.
-----------------------------
9. Politics: suggest to use Round Table instead
There are important conversations to be had about things like gender diversity in math contests, etc., for sure. However, from experience we think that C&P is historically not a good place to have these conversations, as they go off the rails very quickly. We encourage you to use the Round Table instead, where it is much more clear that all posts need to be serious.
-----------------------------
10. MAA complaints: discouraged
We don't want to pretend that the MAA is perfect or that we agree with everything they do. However, we chose to discourage this sort of behavior because in practice most of the comments we see are not useful and some are frankly offensive.
[list] [*] If you just want to blow off steam, do it on your blog instead.
[*] When you have criticism, it should be reasoned, well-thought and constructive. What we mean by this is, for example, when the AOIME was announced, there was great outrage about potential cheating. Well, do you really think that this is something the organizers didn't think about too? Simply posting that "people will cheat and steal my USAMOO qualification, the MAA are idiots!" is not helpful as it is not bringing any new information to the table.
[*] Even if you do have reasoned, well-thought, constructive criticism, we think it is actually better to email it the MAA instead, rather than post it here. Experience shows that even polite, well-meaning suggestions posted in C&P are often derailed by less mature users who insist on complaining about everything.
[/list]
-----------------------------
11. Memes and joke posts: discouraged
It's fine to make jokes or lighthearted posts every so often. But it should be done with discretion. Ideally, jokes should be done within a longer post that has other content. For example, in my response to one user's question about olympiad combinatorics, I used a silly picture of Sogiita Gunha, but it was done within a context of a much longer post where it was meant to actually make a point.

On the other hand, there are many threads which consist largely of posts whose only content is an attached meme with the word "MAA" in it. When done in excess like this, the jokes reflect poorly on the community, so we explicitly discourage them.
-----------------------------
12. Questions that no one can answer: discouraged
Examples of this: "will MIT ask for AOIME scores?", "what will the AIME 2021 cutoffs be (asked in 2020)", etc. Basically, if you ask a question on this forum, it's better if the question is something that a user can plausibly answer :)
-----------------------------
13. Blind speculation: discouraged
Along these lines, if you do see a question that you don't have an answer to, we discourage "blindly guessing" as it leads to spreading of baseless rumors. For example, if you see some user posting "why are there fewer qualifiers than usual this year?", you should not reply "the MAA must have been worried about online cheating so they took fewer people!!". Was sich überhaupt sagen lässt, lässt sich klar sagen; und wovon man nicht reden kann, darüber muss man schweigen.
-----------------------------
14. Discussion of cheating: strongly discouraged
If you have evidence or reasonable suspicion of cheating, please report this to your Competition Manager or to the AMC HQ; these forums cannot help you.
Otherwise, please avoid public discussion of cheating. That is: no discussion of methods of cheating, no speculation about how cheating affects cutoffs, and so on --- it is not helpful to anyone, and it creates a sour atmosphere. A longer explanation is given in Seriously, please stop discussing how to cheat.
-----------------------------
15. Cutoff jokes: never allowed
Whenever the cutoffs for any major contest are released, it is very obvious when they are official. In the past, this has been achieved by the numbers being posted on the official AMC website (here) or through a post from the AMCDirector account.

You must never post fake cutoffs, even as a joke. You should also refrain from posting cutoffs that you've heard of via email, etc., because it is better to wait for the obvious official announcement. A longer explanation is given in A Treatise on Cutoff Trolling.
-----------------------------
16. Meanness: never allowed
Being mean is worse than being immature and unproductive. If another user does something which you think is inappropriate, use the Report button to bring the post to moderator attention, or if you really must reply, do so in a way that is tactful and constructive rather than inflammatory.
-----------------------------

Finally, we remind you all to sit back and enjoy the problems. :D

-----------------------------
(EDIT 2024-09-13: AoPS has asked to me to add the following item.)

Advertising paid program or service: never allowed

Per the AoPS Terms of Service (rule 5h), general advertisements are not allowed.

While we do allow advertisements of official contests (at the MAA and MATHCOUNTS level) and those run by college students with at least one successful year, any and all advertisements of a paid service or program is not allowed and will be deleted.
0 replies
v_Enhance
Jun 12, 2020
0 replies
k i Stop looking for the "right" training
v_Enhance   50
N Oct 16, 2017 by blawho12
Source: Contest advice
EDIT 2019-02-01: https://blog.evanchen.cc/2019/01/31/math-contest-platitudes-v3/ is the updated version of this.

EDIT 2021-06-09: see also https://web.evanchen.cc/faq-contest.html.

Original 2013 post
50 replies
v_Enhance
Feb 15, 2013
blawho12
Oct 16, 2017
lil trip to ancient egypt
ChuMath   11
N 35 minutes ago by Craftybutterfly
Source: 2025 AMC 8 Problem #2
The table below shows the ancient Egyptian hieroglyphs that were used to represent different numbers.

(need asymptote)

For example, the number 32 was represented by (need again). What number was represented by the following combination of hieroglyphs?

(and once again)

$\textbf{(A) } 1,423\qquad\textbf{(B) } 10,423\qquad\textbf{(C) } 14,023\qquad\textbf{(D) } 14,203\qquad\textbf{(E) } 14,230$

my bad @sillysharky
11 replies
ChuMath
Jan 30, 2025
Craftybutterfly
35 minutes ago
Number theory question with many (confusing) variables
urfinalopp   1
N 40 minutes ago by mathprodigy2011
Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
1 reply
urfinalopp
3 hours ago
mathprodigy2011
40 minutes ago
FB = BK , circumcircle and altitude related (In the World of Mathematics 516)
parmenides51   4
N an hour ago by Nioronean
Let $BT$ be the altitude and $H$ be the intersection point of the altitudes of triangle $ABC$. Point $N$ is symmetric to $H$ with respect to $BC$. The circumcircle of triangle $ATN$ intersects $BC$ at points $F$ and $K$. Prove that $FB = BK$.

(V. Starodub, Kyiv)
4 replies
parmenides51
Apr 19, 2020
Nioronean
an hour ago
USAJMO #5 - points on a circle
hrithikguy   205
N an hour ago by cappucher
Points $A,B,C,D,E$ lie on a circle $\omega$ and point $P$ lies outside the circle. The given points are such that (i) lines $PB$ and $PD$ are tangent to $\omega$, (ii) $P, A, C$ are collinear, and (iii) $DE \parallel AC$. Prove that $BE$ bisects $AC$.
205 replies
hrithikguy
Apr 28, 2011
cappucher
an hour ago
BOMBARDIRO CROCODILO VS TRALALERO TRALALA
LostDreams   48
N an hour ago by OmenOrNot
Source: USAJMO 2025/4
Let $n$ be a positive integer, and let $a_0,\,a_1,\dots,\,a_n$ be nonnegative integers such that $a_0\ge a_1\ge \dots\ge a_n.$ Prove that
\[
\sum_{i=0}^n i\binom{a_i}{2}\le\frac{1}{2}\binom{a_0+a_1+\dots+a_n}{2}.
\]Note: $\binom{k}{2}=\frac{k(k-1)}{2}$ for all nonnegative integers $k$.
48 replies
LostDreams
Yesterday at 12:11 PM
OmenOrNot
an hour ago
Tidal wave jumpscare
centslordm   28
N 2 hours ago by aimestew
Source: 2024 AMC 12A #20
Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\triangle ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$

$\textbf{(A) } \left[\frac 38, \frac 12\right] \qquad \textbf{(B) } \left(\frac 12, \frac 23\right] \qquad \textbf{(C) } \left(\frac 23, \frac 34\right] \qquad \textbf{(D) } \left(\frac 34, \frac 78\right] \qquad \textbf{(E) } \left(\frac 78, 1\right]$
28 replies
centslordm
Nov 7, 2024
aimestew
2 hours ago
AMC 10.........
BAM10   17
N 2 hours ago by jkim0656
I'm in 8th grade and have never taken the AMC 10. I am currently in alg2. I have scored 20 on AMC 8 this year and 34 on the chapter math counts last year. Can I qualify for AIME. Also what should I practice AMC 10 next year?
17 replies
BAM10
Mar 2, 2025
jkim0656
2 hours ago
Scary Binomial Coefficient Sum
EpicBird08   32
N 2 hours ago by plang2008
Source: USAMO 2025/5
Determine, with proof, all positive integers $k$ such that $$\frac{1}{n+1} \sum_{i=0}^n \binom{n}{i}^k$$is an integer for every positive integer $n.$
32 replies
1 viewing
EpicBird08
Yesterday at 11:59 AM
plang2008
2 hours ago
0 on jmo
Rong0625   42
N 2 hours ago by llddmmtt1
How many people actually get a flat 0/42 on jmo? I took it for the first time this year and I had never done oly math before so I really only had 2 weeks to figure it out since I didn’t think I would qual. I went in not expecting much but I didn’t think I wouldn’t be able to get ANYTHING. So I’m pretty sure I got 0/42 (unless i get pity points for writing incorrect solutions). Is that bad, am I sped, and should I be embarrassed? Or do other people actually also get 0?
42 replies
Rong0625
Yesterday at 12:14 PM
llddmmtt1
2 hours ago
funny title placeholder
pikapika007   50
N 2 hours ago by llddmmtt1
Source: USAJMO 2025/6
Let $S$ be a set of integers with the following properties:
[list]
[*] $\{ 1, 2, \dots, 2025 \} \subseteq S$.
[*] If $a, b \in S$ and $\gcd(a, b) = 1$, then $ab \in S$.
[*] If for some $s \in S$, $s + 1$ is composite, then all positive divisors of $s + 1$ are in $S$.
[/list]
Prove that $S$ contains all positive integers.
50 replies
+1 w
pikapika007
Yesterday at 12:10 PM
llddmmtt1
2 hours ago
Day Before Tips
elasticwealth   74
N 2 hours ago by awesometriangles
Hi Everyone,

USA(J)MO is tomorrow. I am a Junior, so this is my last chance. I made USAMO by ZERO points but I've actually been studying oly seriously since JMO last year. I am more stressed than I was before AMC/AIME because I feel Olympiad is more unpredictable and harder to prepare for. I am fairly confident in my ability to solve 1/4 but whether I can solve the rest really leans on the topic distribution.

Anyway, I'm just super stressed and not sure what to do. All tips are welcome!

Thanks everyone! Good luck tomorrow!
74 replies
elasticwealth
Mar 19, 2025
awesometriangles
2 hours ago
Inequalities
sqing   3
N 3 hours ago by sqing
Let $ a,b> 0$ and $ a+b=1 . $ Prove that
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{4+\frac{k}{4096}}{1+ ka^7b^7}$$Where $\frac{8192}{3}\geq  k>0 .$
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{\frac{14}{3}}{1+ \frac{8192}{3}a^7b^7}$$
3 replies
sqing
4 hours ago
sqing
3 hours ago
Inequalities
sqing   0
3 hours ago
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that$$a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$$Equality holds when $ a=b=c=\frac{1}{3} $ or $ a=0,b=\frac{3}{4},c=\frac{1}{4} $ or $ a=\frac{1}{4} ,b=0,c=\frac{3}{4} $
or $ a=\frac{3}{4} ,b=\frac{1}{4},c=0. $
0 replies
sqing
3 hours ago
0 replies
combo j3 :blobheart:
rhydon516   21
N 5 hours ago by CatinoBarbaraCombinatoric
Source: USAJMO 2025/3
Let $m$ and $n$ be positive integers, and let $\mathcal R$ be a $2m\times 2n$ grid of unit squares.

A domino is a $1\times2$ or $2\times1$ rectangle. A subset $S$ of grid squares in $\mathcal R$ is domino-tileable if dominoes can be placed to cover every square of $S$ exactly once with no domino extending outside of $S$. Note: The empty set is domino tileable.

An up-right path is a path from the lower-left corner of $\mathcal R$ to the upper-right corner of $\mathcal R$ formed by exactly $2m+2n$ edges of the grid squares.

Determine, with proof, in terms of $m$ and $n$, the number of up-right paths that divide $\mathcal R$ into two domino-tileable subsets.
21 replies
rhydon516
Mar 20, 2025
CatinoBarbaraCombinatoric
5 hours ago
Inequalites
Mario16   17
N Mar 17, 2025 by sqing
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
17 replies
Mario16
Feb 1, 2021
sqing
Mar 17, 2025
Inequalites
G H J
G H BBookmark kLocked kLocked NReply
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Mario16
100 posts
#1 • 1 Y
Y by Mango247
If a+b+c=3 ;a,b,c>=0 prove that 1/(5+a^2)+1/(5+b^2)+1/(5+c^2)<=1/2
Z K Y
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IAmTheHazard
5000 posts
#2
Y by
You literally posted this 2 hours ago.
Z K Y
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Mario16
100 posts
#3
Y by
Yes but i forgot to write Something
Z K Y
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IAmTheHazard
5000 posts
#4
Y by
It seems to be the exact same to me.
Z K Y
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Wildabandon
507 posts
#5
Y by
Mario16 wrote:
Yes but i forgot to write Something

You can edit your post.

If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
This post has been edited 1 time. Last edited by Wildabandon, Feb 2, 2021, 12:23 AM
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Quantum_fluctuations
1282 posts
#6 • 1 Y
Y by Mango247
This is where you should go.
https://artofproblemsolving.com/community/c6h2387664
This post has been edited 1 time. Last edited by Quantum_fluctuations, Feb 2, 2021, 12:23 AM
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KP9
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#7
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Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

So we have : $\sum \frac{1}{5+a^2} \leq \frac{3}{5} - \frac{1}{90}(a^3+b^3+c^3) - \frac{1}{45}(a+b+c) = \frac{3}{5} -\frac{3}{45} - \frac{1}{90}(a^3+b^3+c^3) = \frac{8}{15} - \frac{a^3+b^3+c^3}{90}$ (1)

We also have : $(1+1+1)(1+1+1)(a^3+b^3+c^3)\geq (a+b+c)^3$

$\Rightarrow a^3 + b^3 + c^3 \geq 3$ (2)

(1) and (2) $\Rightarrow \sum \frac{1}{5+a^2} \leq \frac{8}{15} - \frac{3}{90} = \frac{1}{2}$ ( Q.E.D)
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Quantum_fluctuations
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#8
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KP9 wrote:
Another solution :

Easy to prove : $\frac{1}{5+a^2} \leq \frac{1}{5}(1-\frac{1}{18}a^3 - \frac{1}{9}a)$

How did you find that?
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KP9
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#10 • 1 Y
Y by Mango247
oh , iam sorry , i have a problem when i prove it
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logrange
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#11 • 1 Y
Y by Mango247
I have another similar problem in which sum is cyclic in one variable only. In these kind of problems we can use tangent line, but I want to know that whether it can be used in problems which have expression ≤ constant. I have used it only in cases like expression ≥ constant. If you can do this by tangent line then please post the solution of this by tangent line also.
Prove that cyclic sum $\frac{a}{2a^2+a+1}\leq \frac{3}{4}$
Given a+b+c=3 (Sorry, I missed that earlier)
This post has been edited 3 times. Last edited by logrange, Feb 2, 2021, 5:39 PM
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logrange
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#12
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Bump bump bump
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logrange
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#13
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Anyone?
Note - I want a solution without n-1 EV (Calculus)
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Wildabandon
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#14
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I'm thinking Jensen but still using the calculus LOL
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logrange
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#15
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@below
Thanks
This post has been edited 1 time. Last edited by logrange, Feb 3, 2021, 11:13 AM
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starchan
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#16
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I think Chebyshev's kills this one..
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sqing
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#17
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Wildabandon wrote:
If $a,b,c\ge 0$ and $a+b+c=3$, prove that
\[\frac{1}{5+a^2} + \frac{1}{5+b^2} + \frac{1}{5+c^2}\le \frac{1}{2}\]
https://artofproblemsolving.com/community/c4h2391785p19635446
https://artofproblemsolving.com/community/c6h2387664p20486022
Let $a,b,c$ be non-negative numbers such that $ab+bc+ca+abc=4.$ Prove that
$$\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}= 1$$$$\frac{1}{2}\leq \frac{1}{a^2+4}+\frac{1}{b^2+4}+\frac{1}{c^2+4}\leq \frac{3}{5}$$https://artofproblemsolving.com/community/c6h1510436p8962718
Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$\frac{1}{a^2+5}+\frac{1}{b^2+5}+\frac{1}{c^2+5}\leq \frac{1}{2}$$( Vasile Cîrtoaje)
$$a^2b+b^2c+c^2a\leq \frac{368}{3}-\frac{176\sqrt{33}}{9}$$https://artofproblemsolving.com/community/c6h382474p2119615
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 2:55 PM
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sqing
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#18
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Let $ a,b,c>0 $ and $a^2+b^2+c^2+ab+bc+ca =6.$ Prove that
$$ ab+bc+ca-  abc\leq 2$$$$\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\geq \frac{3}{2}$$$$\frac{1}{a^2+8}+\frac{1}{b^2+8}+\frac{1}{c^2+8}\leq \frac{1}{3}$$
This post has been edited 2 times. Last edited by sqing, Mar 17, 2025, 3:30 PM
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sqing
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#19
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Let $ a,b\geq 0 $ and $a+b+a^2+ab+b^2 =5.$ Prove that
$$ \frac{1}{a^2+1}+ \frac{1}{b^2+1}  \geq1$$$$  \frac{1}{a^2+2}+ \frac{1}{b^2+2} \geq \frac{2}{3}$$$$  \frac{1}{a^2+\frac{53}{20}}+ \frac{1}{b^2+\frac{53}{20}} \geq \frac{40}{73}$$$$  \frac{1}{a^2+\frac{1327}{500}}+ \frac{1}{b^2+ \frac{1327}{500}} \geq \frac{1000}{1827}$$$$ \frac{1}{a^2+3}+ \frac{1}{b^2+3}  \geq \frac{185+3\sqrt{21}}{402}$$
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