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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.

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[*]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 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
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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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All the numbers to be zero after finitely many operations
orl   9
N 16 minutes ago by User210790
Source: IMO Shortlist 1989, Problem 19, ILL 64
A natural number is written in each square of an $ m \times n$ chess board. The allowed move is to add an integer $ k$ to each of two adjacent numbers in such a way that non-negative numbers are obtained. (Two squares are adjacent if they have a common side.) Find a necessary and sufficient condition for it to be possible for all the numbers to be zero after finitely many operations.
9 replies
orl
Sep 18, 2008
User210790
16 minutes ago
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A number theory problem
super1978   2
N 32 minutes ago by Tintarn
Source: Somewhere
Let $a,b,n$ be positive integers such that $\sqrt[n]{a}+\sqrt[n]{b}$ is an integer. Prove that $a,b$ are both the $n$th power of $2$ positive integers.
2 replies
super1978
May 11, 2025
Tintarn
32 minutes ago
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Mega angle chase
kjhgyuio   2
N an hour ago by Jupiterballs
Source: https://mrdrapermaths.wordpress.com/2021/01/30/filtering-with-basic-angle-facts/
........
2 replies
kjhgyuio
3 hours ago
Jupiterballs
an hour ago
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power of a point
BekzodMarupov   0
an hour ago
Source: lemmas in olympiad geometry
Epsilon 1.3. Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
0 replies
BekzodMarupov
an hour ago
0 replies
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Interesting inequalities
sqing   1
N 2 hours ago by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ab+bc+ca- abc =3.$ Show that
$$a+k(b+c)\geq 2\sqrt{3 k}$$Where $ k\geq 1. $
Let $a,b,c \geq 0 $ and $2(ab+bc+ca)- abc =31.$ Show that
$$a+k(b+c)\geq \sqrt{62k}$$Where $ k\geq 1. $
1 reply
sqing
2 hours ago
sqing
2 hours ago
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euler function
mathsearcher   0
2 hours ago
Prove that there exists infinitely many positive integers n such that
ϕ(n) | n+1
0 replies
mathsearcher
2 hours ago
0 replies
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Simple but hard
Lukariman   1
N 3 hours ago by Giant_PT
Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
1 reply
Lukariman
4 hours ago
Giant_PT
3 hours ago
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Floor function and coprime
mofumofu   13
N 3 hours ago by Thapakazi
Source: 2018 China TST 2 Day 2 Q4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
13 replies
mofumofu
Jan 9, 2018
Thapakazi
3 hours ago
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Old problem
kwin   0
3 hours ago
Let $ a, b, c > 0$ . Prove that:
$$(a^2+b^2)(b^2+c^2)(c^2+a^2)(ab+bc+ca)^2 \ge 8(abc)^2(a^2+b^2+c^2)^2$$
0 replies
1 viewing
kwin
3 hours ago
0 replies
J
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Interesting inequalities
sqing   2
N 3 hours ago by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ abc+2(ab+bc+ca) =32.$ Show that
$$ka+b+c\geq 8\sqrt k-2k$$Where $0<k\leq 4. $
$$ka+b+c\geq 8 $$Where $ k\geq 4. $
$$a+b+c\geq 6$$$$2a+b+c\geq 8\sqrt 2-4$$
2 replies
sqing
Yesterday at 2:51 PM
sqing
3 hours ago
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RMM 2013 Problem 3
dr_Civot   79
N 3 hours ago by Ilikeminecraft
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$. The lines $AB$ and $CD$ meet at $P$, the lines $AD$ and $BC$ meet at $Q$, and the diagonals $AC$ and $BD$ meet at $R$. Let $M$ be the midpoint of the segment $PQ$, and let $K$ be the common point of the segment $MR$ and the circle $\omega$. Prove that the circumcircle of the triangle $KPQ$ and $\omega$ are tangent to one another.
79 replies
dr_Civot
Mar 2, 2013
Ilikeminecraft
3 hours ago
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Inspired by KhuongTrang
sqing   7
N 4 hours ago by TNKT
Source: Own
Let $a,b,c\ge 0 $ and $ a+b+c=3.$ Prove that
$$\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{6}{abc+5}$$$$\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{7}{abc+6}$$$$\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{21}{17(abc+1)}$$$$\frac{1}{ab+2}+\frac{1}{bc+2}+\frac{1}{ca+2}\ge \frac{42}{17(abc+2)}$$
7 replies
sqing
Jan 21, 2024
TNKT
4 hours ago
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Cycle in a graph with a minimal number of chords
GeorgeRP   5
N 4 hours ago by Photaesthesia
Source: Bulgaria IMO TST 2025 P3
In King Arthur's court every knight is friends with at least $d>2$ other knights where friendship is mutual. Prove that King Arthur can place some of his knights around a round table in such a way that every knight is friends with the $2$ people adjacent to him and between them there are at least $\frac{d^2}{10}$ friendships of knights that are not adjacent to each other.
5 replies
GeorgeRP
Wednesday at 7:51 AM
Photaesthesia
4 hours ago
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Sum of bad integers to the power of 2019
mofumofu   8
N 4 hours ago by Orthocenter.THOMAS
Source: China TST 2019 Test 2 Day 2 Q6
Given coprime positive integers $p,q>1$, call all positive integers that cannot be written as $px+qy$(where $x,y$ are non-negative integers) bad, and define $S(p,q)$ to be the sum of all bad numbers raised to the power of $2019$. Prove that there exists a positive integer $n$, such that for any $p,q$ as described, $(p-1)(q-1)$ divides $nS(p,q)$.
8 replies
mofumofu
Mar 11, 2019
Orthocenter.THOMAS
4 hours ago
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nice or hard
VIMF   1
N Nov 20, 2009 by kuing
let $ a,b,c\geq 0$. prove that
$ \frac{4a}{a+b}+\frac{4b}{b+c}+\frac{4c}{c+a} +\frac{ab^2+bc^2+ca^2+abc}{a^2b+b^2c+c^2a+abc}\geq 7$
1 reply
VIMF
Nov 20, 2009
kuing
Nov 20, 2009
J
nice or hard
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Reveal topic tags
inequalities unsolved
inequalities
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VIMF
52 posts
VIMF
#1 h  Nov 20, 2009, 1:29 PM • 2 Y
Y by Adventure10, Mango247
let $ a,b,c\geq 0$. prove that
$ \frac{4a}{a+b}+\frac{4b}{b+c}+\frac{4c}{c+a} +\frac{ab^2+bc^2+ca^2+abc}{a^2b+b^2c+c^2a+abc}\geq 7$
Z K Y
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kuing
1008 posts
kuing
#2 h  Nov 20, 2009, 1:36 PM • 2 Y
Y by Adventure10, Mango247
VIMF wrote:
let $ a,b,c\geq 0$. prove that
$ \frac {4a}{a + b} + \frac {4b}{b + c} + \frac {4c}{c + a} + \frac {ab^2 + bc^2 + ca^2 + abc}{a^2b + b^2c + c^2a + abc}\geq 7$

ha ...

$ \frac {4a}{a + b} + \frac {4b}{b + c} + \frac {4c}{c + a} + \frac {ab^2 + bc^2 + ca^2 + abc}{a^2b + b^2c + c^2a + abc} - 7$ $ = {\frac {\left( a - b\right) ^{2} \left( b -c \right) ^{2} \left( c - a\right) ^{2}}{\left( a + b \right) \left( b + c \right) \left( c + a\right) \left( {a}^{2}b + {b}^{2}c + {c}^{2}a + abc \right) }}$
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