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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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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Hard geometry
Lukariman   9
N 2 minutes ago by Captainscrubz
Given circle (O) and chord AB with different diameters. The tangents of circle (O) at A and B intersect at point P. On the small arc AB, take point C so that triangle CAB is not isosceles. The lines CA and BP intersect at D, BC and AP intersect at E. Prove that the centers of the circles circumscribing triangles ACE, BCD and OPC are collinear.
9 replies
+1 w
Lukariman
May 14, 2025
Captainscrubz
2 minutes ago
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INMO 2007 Problem 6
Sathej   32
N 19 minutes ago by math_genie
Source: Inequality
If $ x$, $ y$, $ z$ are positive real numbers, prove that
\[ \left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .\]
32 replies
Sathej
Feb 4, 2007
math_genie
19 minutes ago
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Prove that the triangle is isosceles.
TUAN2k8   3
N 21 minutes ago by TUAN2k8
Source: My book
Given acute triangle $ABC$ with two altitudes $CF$ and $BE$.Let $D$ be the point on the line $CF$ such that $DB \perp BC$.The lines $AD$ and $EF$ intersect at point $X$, and $Y$ is the point on segment $BX$ such that $CY \perp BY$.Suppose that $CF$ bisects $BE$.Prove that triangle $ACY$ is isosceles.
3 replies
TUAN2k8
2 hours ago
TUAN2k8
21 minutes ago
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Choice of indexes in a sequence
randomusername   2
N 33 minutes ago by EthanWYX2009
Source: Kürschák 2005, problem 1
Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$. Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that
\[\sum_{i=1}^k n_ia_{n_{i-1}}<2005.\]
2 replies
randomusername
Jul 13, 2014
EthanWYX2009
33 minutes ago
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Simple but hard
Lukariman   3
N 41 minutes 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.
3 replies
Lukariman
Today at 2:47 AM
Giant_PT
41 minutes ago
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Ah, easy one
irregular22104   2
N an hour ago by irregular22104
Source: Own
In the number series $1,9,9,9,8,...,$ every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers $1234$ and $5678$ in this series?
2 replies
irregular22104
Wednesday at 4:01 PM
irregular22104
an hour ago
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power of a point
BekzodMarupov   1
N an hour ago by nabodorbuco2
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.
1 reply
BekzodMarupov
Today at 5:41 AM
nabodorbuco2
an hour ago
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Functional Equation!
EthanWYX2009   5
N an hour ago by Miquel-point
Source: 2025 TST 24
Find all functions $f:\mathbb Z\to\mathbb Z$ such that $f$ is unbounded and
\[2f(m)f(n)-f(n-m)-1\]is a perfect square for all integer $m,n.$
5 replies
EthanWYX2009
Mar 29, 2025
Miquel-point
an hour ago
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IMO Shortlist 2014 G3
hajimbrak   46
N an hour ago by Rayvhs
Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$.
(Here we always assume that an angle bisector is a ray.)

Proposed by Sergey Berlov, Russia
46 replies
hajimbrak
Jul 11, 2015
Rayvhs
an hour ago
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Eight-point cicle
sandu2508   15
N an hour ago by Mamadi
Source: Balkan MO 2010, Problem 2
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
15 replies
sandu2508
May 4, 2010
Mamadi
an hour ago
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IMO Solution mistake
CHESSR1DER   1
N 2 hours ago by whwlqkd
Source: Mistake in IMO 1982/1 4th solution
I found a mistake in 4th solution at IMO 1982/1. It gives answer $660$ and $661$. But right answer is only $660$. Should it be reported somewhere in Aops?
1 reply
CHESSR1DER
5 hours ago
whwlqkd
2 hours ago
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inequality
mathematical-forest   2
N 2 hours ago by mathematical-forest
For positive real intengers $x_{1} ,x_{2} ,\cdots,x_{n} $, such that $\prod_{i=1}^{n} x_{i} =1$
proof:
$$\sum_{i=1}^{n} \frac{1}{1+\sum _{j\ne i}x_{j} } \le 1$$
2 replies
mathematical-forest
Yesterday at 12:40 PM
mathematical-forest
2 hours ago
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4-vars inequality
xytunghoanh   4
N 2 hours ago by JARP091
For $a,b,c,d \ge 0$ and $a\ge c$, $b \ge d$. Prove that
$$a+b+c+d+ac+bd+8 \ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{cd}+\sqrt{da}+\sqrt{ac}+\sqrt{bd})$$.
4 replies
xytunghoanh
Yesterday at 2:10 PM
JARP091
2 hours ago
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Sneaky one
Sunjee   2
N 2 hours ago by Sunjee
Find minimum and maximum value of following function.
$$f(x,y)=\frac{\sqrt{x^2+y^2}+\sqrt{(x-2)^2+(y-1)^2}}{\sqrt{x^2+(y-1)^2}+\sqrt{(x-2)^2+y^2}} $$
2 replies
Sunjee
3 hours ago
Sunjee
2 hours ago
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Inspired by old results
sqing   1
N Apr 16, 2025 by sqing
Source: Own
Let $ a,b \geq 0 $ and $ a^2+b^2+a+b \geq 4 .$ Prove that$$ \frac{1}{a^2+b+1}+\frac{1}{b^2+a+1}+\frac{1}{a+b+1} \leq \frac{7\sqrt{17}-1}{26}$$
1 reply
sqing
Apr 16, 2025
sqing
Apr 16, 2025
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Inspired by old results
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Reveal topic tags
inequalities proposed
algebra
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sqing
42191 posts
sqing
#1 h  Apr 16, 2025, 2:44 PM • 1 Y
Y by cubres
Let $ a,b \geq 0 $ and $ a^2+b^2+a+b \geq 4 .$ Prove that$$ \frac{1}{a^2+b+1}+\frac{1}{b^2+a+1}+\frac{1}{a+b+1} \leq \frac{7\sqrt{17}-1}{26}$$
This post has been edited 2 times. Last edited by sqing, Apr 16, 2025, 3:00 PM
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sqing
42191 posts
sqing
#2 h  Apr 16, 2025, 2:58 PM • 1 Y
Y by cubres
Let $ a,b \geq 0 $ and $ a^2+b^2+ab+a+b\geq 5 .$ Prove that$$ \frac{a}{2a+b^2}+\frac{b}{2b+1}+\frac{1}{a^2+2} \leq 1 $$
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