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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.

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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.
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0 replies
jlacosta
Mar 2, 2025
0 replies
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k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
J
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k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
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keep one card and discard the other
Scilyse   2
N 34 minutes ago by flower417477
Source: CGMO 2024 P2
There are $8$ cards on which the numbers $1$, $2$, $\dots$, $8$ are written respectively. Alice and Bob play the following game: in each turn, Alice gives two cards to Bob, who must keep one card and discard the other. The game proceeds for four turns in total; in the first two turns, Bob cannot keep both of the cards with the larger numbers, and in the last two turns, Bob also cannot keep both of the cards with the larger numbers. Let $S$ be the sum of the numbers written on the cards that Bob keeps. Find the greatest positive integer $N$ for which Bob can guarantee that $S$ is at least $N$.
2 replies
Scilyse
Jan 28, 2025
flower417477
34 minutes ago
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JBMO Shortlist 2020 N4
Lukaluce   6
N 34 minutes ago by Assassino9931
Source: JBMO Shortlist 2020
Find all prime numbers $p$ such that

$(x + y)^{19} - x^{19} - y^{19}$

is a multiple of $p$ for any positive integers $x$, $y$.
6 replies
Lukaluce
Jul 4, 2021
Assassino9931
34 minutes ago
J
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Existence in number theory
shangyang   5
N an hour ago by shanelin-sigma
Prove that there are infinitely many integers can't be written as $$\frac{p^a-p^b}{p^c-p^d}$$, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.
5 replies
shangyang
Nov 26, 2021
shanelin-sigma
an hour ago
J
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Interesting inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 2.$ Prove that
$$ (a+1)(b+1)(c +1)-\frac{9}{4}abc\leq 9$$$$ (a+2)(b+2)(c +2)-4 abc\leq 32$$$$ (a+2)(b+2)(c +2)-\frac{17}{4}a b c\leq 30$$$$ (a+1)(b+1)(c +1)-\frac{23}{10}abc\leq\frac{43}{5}$$
1 reply
sqing
3 hours ago
sqing
an hour ago
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All Russian Olympiad 2018 Day1 P2
Davrbek   23
N 2 hours ago by Marcus_Zhang
Source: Grade 11 P2
Let $n\geq 2$ and $x_{1},x_{2},\ldots,x_{n}$ positive real numbers. Prove that
\[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]
23 replies
Davrbek
Apr 28, 2018
Marcus_Zhang
2 hours ago
J
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Interesting inequality
sqing   1
N 2 hours ago by sqing
Source: Own
Let $ a,b,c\geq \frac{1}{3}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 17+2\sqrt{73}$$Let $ a,b,c\geq \frac{1}{2}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{469+115\sqrt{17}}{32}$$Let $ a,b,c\geq \frac{1}{5}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq \frac{569+34\sqrt{281}}{25}$$
1 reply
sqing
2 hours ago
sqing
2 hours ago
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Inequalities
sqing   4
N 2 hours ago by sqing
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
4 replies
sqing
Wednesday at 1:42 PM
sqing
2 hours ago
J
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True Generalization of 2023 CGMO T7
EthanWYX2009   0
2 hours ago
Source: aops.com/community/c6h3132846p28384612
Given positive integer $n.$ Let $x_1,\ldots ,x_n\ge 0$ and $x_1x_2\cdots x_n\le 1.$ Show that
\[\sum_{k=1}^n\frac{1}{1+\sum_{j\neq k}x_j}\le\frac n{1+(n-1)\sqrt[n]{x_1x_2\cdots x_n}}.\]
0 replies
1 viewing
EthanWYX2009
2 hours ago
0 replies
J
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Not homogenous, messy inequality
Kimchiks926   10
N 2 hours ago by Marcus_Zhang
Source: Latvian TST for Baltic Way 2019 Problem 1
Prove that for all positive real numbers $a, b, c$ with $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} =1$ the following inequality holds:
$$3(ab+bc+ca)+\frac{9}{a+b+c} \le \frac{9abc}{a+b+c} + 2(a^2+b^2+c^2)+1$$
10 replies
Kimchiks926
May 29, 2020
Marcus_Zhang
2 hours ago
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USA 97 [1/(b^3+c^3+abc) + ... >= 1/(abc)]
Maverick   45
N 4 hours ago by Marcus_Zhang
Source: USAMO 1997/5; also: ineq E2.37 in Book: Inegalitati; Authors:L.Panaitopol,V. Bandila,M.Lascu
Prove that, for all positive real numbers $ a$, $ b$, $ c$, the inequality
\[ \frac {1}{a^3 + b^3 + abc} + \frac {1}{b^3 + c^3 + abc} + \frac {1}{c^3 + a^3 + abc} \leq \frac {1}{abc} \]
holds.
45 replies
Maverick
Sep 12, 2003
Marcus_Zhang
4 hours ago
J
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The prime inequality learning problem
orl   137
N 4 hours ago by Marcus_Zhang
Source: IMO 1995, Problem 2, Day 1, IMO Shortlist 1995, A1
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc = 1$. Prove that
\[ \frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}. \]
137 replies
orl
Nov 9, 2005
Marcus_Zhang
4 hours ago
J
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Trig Identity
mithu542   4
N Yesterday at 10:19 PM by franklin2013
Prove the following identity:
$$\dfrac{1+\tan \theta}{1-\tan \theta}=\tan \left(\theta+\dfrac{\pi}{4}\right),$$where $\theta$ is in radians.
4 replies
mithu542
Yesterday at 4:09 PM
franklin2013
Yesterday at 10:19 PM
J
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Prove equation
nhathhuyyp5c   1
N Yesterday at 10:00 PM by no_room_for_error
Let $a,b,c,d$ be distinct reals such that $$(a^2+b^2-1)(a+b)=(b^2+c^2-1)(b+c)=(c^2+d^2-1)(c+d)$$Prove that $a+b+c+d=0$
1 reply
nhathhuyyp5c
Yesterday at 4:18 PM
no_room_for_error
Yesterday at 10:00 PM
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FB = BK , circumcircle and altitude related (In the World of Mathematics 516)
parmenides51   2
N Yesterday at 9:59 PM by lolsamo
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)
2 replies
parmenides51
Apr 19, 2020
lolsamo
Yesterday at 9:59 PM
J
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J
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Inequality with integers and indices
Michael Niland   7
N Mar 18, 2025 by Ninjametry
Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$, and for a positive integer $n\geq3$, prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$.
7 replies
Michael Niland
Mar 17, 2025
Ninjametry
Mar 18, 2025
J
Inequality with integers and indices
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Reveal topic tags
inequalities
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Michael Niland
681 posts
Michael Niland
#1 h  Mar 17, 2025, 6:52 AM
Y by
Without using a calculator, show that $2^{\frac{1}{2}}$ $< 3^{\frac{1}{3}}$, and for a positive integer $n\geq3$, prove that

$n^{\frac{1}{n}}$ $>(n+1)^{\frac{1}{n+1}}$.
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invisibleman
13 posts
invisibleman
#2 h  Mar 17, 2025, 1:09 PM
Y by
In order to solve the first problem, I have the following idea: raise both sides to the 6th power, and you're done! Right?
In the general case, consider the function f(x)=lnx/x, whose derivative is negative for x>e, so the function is decreasing. Let x=n and x=n+1, and compare them! I think you understand, if not, write and I'll answer!
This post has been edited 3 times. Last edited by invisibleman, Mar 18, 2025, 11:36 AM
Reason: typo
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krish6_9
18 posts
krish6_9
#3 h  Mar 17, 2025, 1:41 PM
Y by
The inequality is equivalent to $n^{n+1} > (n+1)^n$. Taking natural logarithm on both sides, this is equivalent to $(n+1) \ln n > n \ln (n+1).$ Dividing, this is equivalent to $\frac{n+1}{\ln (n+1)} > \frac{n}{\ln n}$. Since $\frac{x}{\ln x}$ is increasing (obviously linear functions increase more quickly than logarithms, or using the derivative like @above said), we are done by definition.

i hope this is right! :D :ddr:
well someone taught me the solution hehe
This post has been edited 2 times. Last edited by krish6_9, Mar 18, 2025, 12:31 AM
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Speedysolver1
80 posts
Speedysolver1
#4 h  Mar 17, 2025, 1:42 PM
Y by
Help me with a forum
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krish6_9
18 posts
krish6_9
#5 h  Mar 17, 2025, 1:44 PM
Y by
who, me? wut does that mean
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ohiorizzler1434
718 posts
ohiorizzler1434
#6 h  Mar 18, 2025, 1:43 AM
Y by
Bro! Let's show n^(n+1) > (n+1)^n , or (1+1/n)^n < n. Note that (1+1/n)^n < e because this limits upwards towards e. So for n >= 3, this is obviously true as e<3. Bruh! Now that's rizz!
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allisok100
1 post
allisok100
#7 h  Mar 18, 2025, 7:32 AM
Y by
f(x)=\frac{\ln x}{x}\Rightarrow f'(x)=\frac{1-\ln x}{{{x}^{2}}}<0,\forall x>e\Rightarrow f(n)>f(n+1),\forall n\ge 2
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Ninjametry
284 posts
Ninjametry
#8 h  Mar 18, 2025, 12:02 PM
Y by
Graph x^(1/x)
now observe where we get maxima
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