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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
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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Easy P4 combi game with nt flavour
Maths_VC   2
N a minute ago by Assassino9931
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
2 replies
Maths_VC
May 27, 2025
Assassino9931
a minute ago
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2-var inequality
sqing   6
N 7 minutes ago by sqing
Source: Own
Let $ a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq 8$$$$ (a^2+b^2)(a+1)(b+1) \leq 8$$Let $ a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$
6 replies
+1 w
sqing
Yesterday at 1:35 PM
sqing
7 minutes ago
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FE i created on bijective function with x≠y
benjaminchew13   3
N 11 minutes ago by benjaminchew13
Source: own (probably)
Find all bijective functions $f:\mathbb{R}\to \mathbb{R}$ such that $$(x-y)f(x+f(f(y)))=xf(x)+f(y)^{2}$$for all $x,y\in \mathbb{R}$ such that $x\neq y$.
3 replies
benjaminchew13
22 minutes ago
benjaminchew13
11 minutes ago
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Old Inequality
giangtruong13   1
N 16 minutes ago by sqing
Let $a,b,c >0$ and $abc=1$. Prove that: $$ \sqrt{a^2-a+1}+\sqrt{b^2-b+1} +\sqrt{c^2-c+1} \ge a+b+c$$
1 reply
giangtruong13
2 hours ago
sqing
16 minutes ago
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Combo resources
Fly_into_the_sky   2
N 37 minutes ago by Fly_into_the_sky
Ok so i never did combinatorics in my life :oops: and i am willing to be able to do P1/P4 combos (or even more)
So yeah how can i start from scratch?
Remark:i don't want compuational combo resources :noo:
2 replies
Fly_into_the_sky
Yesterday at 5:15 PM
Fly_into_the_sky
37 minutes ago
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A very good problem
JetFire008   1
N 44 minutes ago by JetFire008
Source: Spain 1997 (as claimed by the internet)
There are $n$ identical cars on a circular track. Among all of them, they have just enough gas for one car to complete a lap. Show that there is a car that can complete a lap by collecting gas from the other cars on its way around
Read the bold line carefully as it is easy to misread the problem.
1 reply
JetFire008
an hour ago
JetFire008
44 minutes ago
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P lies on BC
Melid   0
an hour ago
Source: own
In scalene triangle $ABC$, which doesn't have right angle, let $O$ be its circumcenter. Let $H_{1}$ and $H_{2}$ be orthocenters of triangle $ABO$ and $ACO$, respectively. Let $O_{1}$ be circumcenter of triangle $OH_{1}H_{2}$. If circle $ACO_{1}$ and circle $CH_{1}H_{2}$ intersect at $P$ for the second time, prove that $P$ lies on $BC$.
0 replies
Melid
an hour ago
0 replies
J
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Polynomial functional equation
Fishheadtailbody   2
N an hour ago by Fishheadtailbody
Source: MACMO
$P(x)$ is a polynomial with real coefficients such that
\[ P(x)^2 - 1 = 4 P(x^2 - 4x + 1). \]Find $P(x)$.

fixed now
2 replies
Fishheadtailbody
Apr 18, 2025
Fishheadtailbody
an hour ago
J
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Strange circles in an orthocenter config
VideoCake   2
N an hour ago by pi_quadrat_sechstel
Source: 2025 German MO, Round 4, Grade 12, P3
Let \(\overline{AD}\) and \(\overline{BE}\) be altitudes in an acute triangle \(ABC\) which meet at \(H\). Suppose that \(DE\) meets the circumcircle of \(ABC\) at \(P\) and \(Q\) such that \(P\) lies on the shorter arc of \(BC\) and \(Q\) lies on the shorter arc of \(CA\). Let \(AQ\) and \(BE\) meet at \(S\). Show that the circumcircles of \(BPE\) and \(QHS\) and the line \(PH\) concur.
2 replies
VideoCake
May 26, 2025
pi_quadrat_sechstel
an hour ago
J
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Lines pass through a common point
April   5
N an hour ago by SatisfiedMagma
Source: Baltic Way 2008, Problem 18
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| = c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.
5 replies
April
Nov 23, 2008
SatisfiedMagma
an hour ago
J
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Parameter and 4 variables
mihaig   1
N 2 hours ago by mihaig
Source: Own
Find the positive real constants $K$ such that
$$3\left(a^2+b^2+c^2+d^2\right)+4\left(abcd\right)^K\geq\left(a+b+c+d\right)^2$$for all $a,b,c,d\geq0$ satisfying $a+b+c+d\geq4.$
1 reply
mihaig
2 hours ago
mihaig
2 hours ago
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How many friends can sit in that circle at most?
Arytva   0
2 hours ago

A group of friends sits in a ring. Each friend picks a different whole number and holds a stone marked with it. Then they pass their stone one seat to the right so everyone ends up with two stones: one they made and one they received. Now they notice something odd: if your original number is $x$, your right-neighbor’s is $y$, and the next person over is $z$, then for every trio in the circle they see

$$ x + z = (2 - x)\,y. $$
They want as many friends as possible before this breaks (since all stones must stay distinct).

How many friends can sit in that circle at most?
0 replies
Arytva
2 hours ago
0 replies
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Reflected point lies on radical axis
Mahdi_Mashayekhi   7
N 2 hours ago by amogususususus
Source: Iran 2025 second round P4
Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.
7 replies
Mahdi_Mashayekhi
Apr 19, 2025
amogususususus
2 hours ago
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Shortlist 2017/G3
fastlikearabbit   124
N 2 hours ago by ND_
Source: Shortlist 2017, Moldova TST 2018
Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.
124 replies
fastlikearabbit
Jul 10, 2018
ND_
2 hours ago
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Inequality
hlminh   1
N Apr 22, 2025 by arqady
Let $a,b,c>0$ such that $a^2+b^2+c^2=3.$ Prove that $\sum \frac a{\sqrt{b^2+b+c}}\leq \sqrt 3.$
1 reply
hlminh
Apr 21, 2025
arqady
Apr 22, 2025
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Inequality
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Reveal topic tags
inequalities
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hlminh
241 posts
hlminh
#1 h  Apr 21, 2025, 9:36 AM
Y by
Let $a,b,c>0$ such that $a^2+b^2+c^2=3.$ Prove that $\sum \frac a{\sqrt{b^2+b+c}}\leq \sqrt 3.$
Z K Y
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arqady
30263 posts
arqady
#2 h  Apr 22, 2025, 1:45 PM
Y by
hlminh wrote:
Let $a,b,c>0$ such that $a^2+b^2+c^2=3.$ Prove that $\sum \frac a{\sqrt{b^2+b+c}}\leq \sqrt 3.$
Are you sure?
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