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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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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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Shortlist 2017/G3
fastlikearabbit   124
N 3 minutes 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
1 viewing
fastlikearabbit
Jul 10, 2018
ND_
3 minutes ago
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Is there a good solution?
sadwinter   0
11 minutes ago
:maybe: :love: :love:
0 replies
sadwinter
11 minutes ago
0 replies
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Factoring 2024 in Z[\sqrt{3}]
Tintarn   2
N 15 minutes ago by Rayholr123
Source: Czech-Polish-Slovak Junior Match 2024, T-2
Among all triples $(a,b,c)$ of natural numbers satisfying
\[(a+14\sqrt{3})(b-14c\sqrt{3})=2024,\]determine the one with the maximal value of $a$.
2 replies
Tintarn
May 29, 2024
Rayholr123
15 minutes ago
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Inspired by old results
sqing   2
N 22 minutes ago by sqing
Source: Own
Let $ a,b> 0. $ Prove that
$$ \frac{a^3}{b^3+ab^2}+ \frac{4b^3}{a^3+b^3+2ab^2}\geq \frac{3}{2}$$$$\frac{a^3}{b^3+(a+b)^3}+ \frac{b^3}{a^3+(a+b)^3}+ \frac{(a+b)^2}{a^2+b^2+ab} \geq \frac{14}{9}$$
2 replies
sqing
Today at 2:55 AM
sqing
22 minutes ago
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Interesting inequality
sqing   4
N 25 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a^2+b^2+c^2 =3.$ Prove that
$$ a^4+ b^4+c^4+6abc\leq9$$$$ a^3+ b^3+ c^3+3( \sqrt{3}-1)abc\leq 3\sqrt 3$$
4 replies
sqing
Yesterday at 2:54 AM
sqing
25 minutes ago
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Parameter and 4 variables
mihaig   0
25 minutes ago
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 $a+b+c+d\geq4.$
0 replies
+1 w
mihaig
25 minutes ago
0 replies
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circumcenter of ARS lies on AD
Melid   0
28 minutes ago
Source: own
In triangle $ABC$, let $D$ be a point on arc $BC$ of circle $ABC$ which doesn't contain $A$. $AD$ and $BC$ intersect at $E$. Let $P$ and $Q$ be the reflection of $E$ about to $AB$ and $AC$, respectively. $PD$ intersects $AB$ at $R$, and $QD$ intersects $AC$ at $S$. Prove that circumcenter of triangle $ARS$ lies on $AD$.
0 replies
Melid
28 minutes ago
0 replies
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A sharp one with 3 var (3)
mihaig   6
N 37 minutes ago by mihaig
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$a^2+b^2+c^2+5abc\geq8.$$
6 replies
mihaig
May 27, 2025
mihaig
37 minutes ago
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geometry problem with many circumcircles
Melid   1
N an hour ago by Beelzebub
Source: own
In scalene triangle $ABC$, which doesn't have right angle, let $O$ be its circumcenter. Circle $BOC$ intersects $AB$ and $AC$ at $A_{1}$ and $A_{2}$ for the second time, respectively. Similarly, circle $COA$ intersects $BC$ and $BA$ at $B_{1}$ and $B_{2}$, and circle $AOB$ intersects $CA$ and $CB$ at $C_{1}$ and $C_{2}$ for the second time, respectively. Let $O_{1}$ and $O_{2}$ be circumcenters of triangle $A_{1}B_{1}C_{1}$ and $A_{2}B_{2}C_{2}$, respectively. Prove that $O, O_{1}, O_{2}$ are collinear.
1 reply
Melid
4 hours ago
Beelzebub
an hour ago
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weird conditions in geo
Davdav1232   3
N an hour ago by lakshya2009
Source: Israel TST 7 2025 p1
Let \( \triangle ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) be a point on \( AC \). Let \( L \) be a point inside the triangle such that \( \angle CLD = 90^\circ \) and
\[ CL \cdot BD = BL \cdot CD. \]Prove that the circumcenter of triangle \( \triangle BDL \) lies on line \( AB \).
3 replies
Davdav1232
May 8, 2025
lakshya2009
an hour ago
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Problem 4
codyj   88
N 2 hours ago by ND_
Source: IMO 2015 #4
Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Proposed by Greece
88 replies
codyj
Jul 11, 2015
ND_
2 hours ago
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pairs (m, n) such that a fractional expression is an integer
cielblue   3
N 2 hours ago by Pal702004
Find all pairs $(m,\ n)$ of positive integers such that $\frac{m^3-mn+1}{m^2+mn+2}$ is an integer.
3 replies
cielblue
May 24, 2025
Pal702004
2 hours ago
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interesting geometry config (3/3)
Royal_mhyasd   0
3 hours ago
Let $\triangle ABC$ be an acute triangle, $H$ its orthocenter and $E$ the center of its nine point circle. Let $P$ be a point on the parallel through $C$ to $AB$ such that $\angle CPH = |\angle BAC-\angle ABC|$ and $P$ and $A$ are on different sides of $BC$ and $Q$ a point on the parallel through $B$ to $AC$ such that $\angle BQH = |\angle BAC - \angle ACB|$ and $C$ and $Q$ are on different sides of $AB$. If $B'$ and $C'$ are the reflections of $H$ over $AC$ and $AB$ respectively, $S$ and $T$ are the intersections of $B'Q$ and $C'P$ respectively with the circumcircle of $\triangle ABC$, prove that the intersection of lines $CT$ and $BS$ lies on $HE$.

final problem for this "points on parallels forming strange angles with the orthocenter" config, for now. personally i think its pretty cool :D
0 replies
Royal_mhyasd
3 hours ago
0 replies
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A perverse one
darij grinberg   7
N 3 hours ago by ezpotd
Source: German TST 2004, IMO ShortList 2003, number problem 2
Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$:

(i) move the last digit of $a$ to the first position to obtain the numb er $b$;
(ii) square $b$ to obtain the number $c$;
(iii) move the first digit of $c$ to the end to obtain the number $d$.

(All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.)

Find all numbers $a$ for which $d\left( a\right) =a^2$.

Proposed by Zoran Sunic, USA
7 replies
darij grinberg
May 18, 2004
ezpotd
3 hours ago
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Number Theory
TUAN2k8   4
N Apr 19, 2025 by vsamc
Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
4 replies
TUAN2k8
Apr 18, 2025
vsamc
Apr 19, 2025
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Number Theory
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fun nt
number theory
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TUAN2k8
39 posts
TUAN2k8
#1 h  Apr 18, 2025, 3:25 PM
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Prove that there are infinitely many of positive integers m such that $m+1 | 3^m+1$
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Roger.Moore
5 posts
Roger.Moore
#2 h  Apr 18, 2025, 6:06 PM
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TUAN2k8
39 posts
TUAN2k8
#3 h  Apr 18, 2025, 11:08 PM
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Any solution?
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epl1
157 posts
epl1
#4 h  Apr 19, 2025, 2:44 AM
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Woah, self made?
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vsamc
3789 posts
vsamc
#5 h  Apr 19, 2025, 2:51 AM • 2 Y
Y by kiyoras_2001, centslordm
$m = 3^{3^k}$ works for all $k\geq 0$, since $$3^{3^k} + 1 \mid (3^{3^k})^{3^{3^k-k}} + 1^{3^{3^k-k}} = 3^{3^{3^k}} + 1.$$
This post has been edited 2 times. Last edited by vsamc, Apr 19, 2025, 2:51 AM
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