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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.

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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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Inequality
SunnyEvan   8
N a few seconds ago by arqady
Let $a$, $b$, $c$ be non-negative real numbers, no two of which are zero. Prove that :
$$ \sum \frac{3ab-2bc+3ca}{3b^2+bc+3c^2} \geq \frac{12}{7}$$
8 replies
SunnyEvan
Apr 1, 2025
arqady
a few seconds ago
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Inequality conjecture
RainbowNeos   2
N 4 minutes ago by RainbowNeos
Show (or deny) that there exists an absolute constant $C>0$ that, for all $n$ and $n$ positive real numbers $x_i ,1\leq i \leq n$, there is
\[\sum_{i=1}^n \frac{x_i^2}{\sum_{j=1}^i x_j}\geq C \ln n\left(\prod_{i=1}^n x_i\right)^{\frac{1}{n}}\]
2 replies
RainbowNeos
May 29, 2025
RainbowNeos
4 minutes ago
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2- player game on a strip of n squares with two game pieces
parmenides51   2
N 6 minutes ago by Gggvds1
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 3
Alice and Bob play a game on a strip of $n \ge 3$ squares with two game pieces. At the beginning, Alice’s piece is on the first square while Bob’s piece is on the last square. The figure shows the starting position for a strip of $ n = 7$ squares.
IMAGE
The players alternate. In each move, they advance their own game piece by one or two squares in the direction of the opponent’s piece. The piece has to land on an empty square without jumping over the opponent’s piece. Alice makes the first move with her own piece. If a player cannot move, they lose.

For which $n$ can Bob ensure a win no matter how Alice plays?
For which $n$ can Alice ensure a win no matter how Bob plays?

(Karl Czakler)
2 replies
parmenides51
Mar 26, 2024
Gggvds1
6 minutes ago
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Incenters and Circles
rkm0959   6
N 10 minutes ago by happypi31415
Source: Korean National Junior Olympiad Problem 1
In a triangle $\triangle ABC$ with incenter $I$,
Let $D$ = $AI$ $\cap$ $BC$
$E$ = incenter of $\triangle ABD$
$F$ = incenter of $\triangle ACD$
$P$ = intersection of $\odot BCE$ and $\overline {ED}$
$Q$ = intersection of $\odot BCF$ and $\overline {FD}$
$M$ = midpoint of $\overline {BC}$

Prove that $D, M, P, Q$ concycle
6 replies
+1 w
rkm0959
Nov 2, 2014
happypi31415
10 minutes ago
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Question on Balkan SL
Fmimch   4
N 28 minutes ago by BreezeCrowd
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
4 replies
Fmimch
Apr 30, 2025
BreezeCrowd
28 minutes ago
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Rays, incircle, angles...
mathisreal   3
N an hour ago by Assassino9931
Source: Rioplatense L-3 2022 #4
Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.
3 replies
mathisreal
Dec 13, 2022
Assassino9931
an hour ago
J
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Wordy Geometry in Taiwan TST
ckliao914   9
N an hour ago by Scilyse
Source: 2023 Taiwan TST Round 3 Mock Exam 6
Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.
9 replies
ckliao914
Apr 29, 2023
Scilyse
an hour ago
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Converse of a classic orthocenter problem
spartacle   43
N 4 hours ago by ihategeo_1969
Source: USA TSTST 2020 Problem 6
Let $A$, $B$, $C$, $D$ be four points such that no three are collinear and $D$ is not the orthocenter of $ABC$. Let $P$, $Q$, $R$ be the orthocenters of $\triangle BCD$, $\triangle CAD$, $\triangle ABD$, respectively. Suppose that the lines $AP$, $BQ$, $CR$ are pairwise distinct and are concurrent. Show that the four points $A$, $B$, $C$, $D$ lie on a circle.

Andrew Gu
43 replies
spartacle
Dec 14, 2020
ihategeo_1969
4 hours ago
J
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Symmetric points part 2
CyclicISLscelesTrapezoid   22
N 4 hours ago by ihategeo_1969
Source: USA TSTST 2022/6
Let $O$ and $H$ be the circumcenter and orthocenter, respectively, of an acute scalene triangle $ABC$. The perpendicular bisector of $\overline{AH}$ intersects $\overline{AB}$ and $\overline{AC}$ at $X_A$ and $Y_A$ respectively. Let $K_A$ denote the intersection of the circumcircles of triangles $OX_AY_A$ and $BOC$ other than $O$.

Define $K_B$ and $K_C$ analogously by repeating this construction two more times. Prove that $K_A$, $K_B$, $K_C$, and $O$ are concyclic.

Hongzhou Lin
22 replies
CyclicISLscelesTrapezoid
Jun 27, 2022
ihategeo_1969
4 hours ago
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Cyclic Quad. and Intersections
Thelink_20   11
N 5 hours ago by americancheeseburger4281
Source: My Problem
Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$. Let $AC\cap BD=E$, $AB\cap CD=F$, $(AEF)\cap\Gamma=X$, $(BEF)\cap\Gamma=Y$, $(CEF)\cap\Gamma=Z$, $(DEF)\cap\Gamma=W$, $XZ\cap YW=M$, $XY\cap ZW=N$. Prove that $MN$ lies over $EF$.
11 replies
Thelink_20
Oct 29, 2024
americancheeseburger4281
5 hours ago
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PE is bisector of BPC
goldeneagle   44
N 5 hours ago by cursed_tangent1434
Source: Iran TST 2012 -first day- problem 2
Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$.

Proposed by Mr.Etesami
44 replies
goldeneagle
Apr 23, 2012
cursed_tangent1434
5 hours ago
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help me~~
Imshyso   1
N Today at 9:09 AM by Royal_mhyasd
Given triangle ABC inscribed in (O) with orthocenter H. Let K be the midpoint of AH. Take E,F on AC, AB so that BKE=CKF=90. Prove that E,O,F are collinear.
1 reply
Imshyso
Today at 8:10 AM
Royal_mhyasd
Today at 9:09 AM
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Isosceles Triangle Geo
oVlad   5
N Today at 8:22 AM by Tamam
Source: Romania Junior TST 2025 Day 1 P2
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$
5 replies
oVlad
Apr 12, 2025
Tamam
Today at 8:22 AM
J
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Easy Geometry
rkm0959   13
N Today at 8:15 AM by Rounak_iitr
Source: 2015 Korean Mathematical Olympiad P2
Let the circumcircle of $\triangle ABC$ be $\omega$. A point $D$ lies on segment $BC$, and $E$ lies on segment $AD$. Let ray $AD \cap \omega = F$. A point $M$, which lies on $\omega$, bisects $AF$ and it is on the other side of $C$ with respect to $AF$. Ray $ME \cap \omega = G$, ray $GD \cap \omega = H$, and $MH \cap AD = K$. Prove that $B, E, C, K$ are cyclic.
13 replies
rkm0959
Nov 1, 2015
Rounak_iitr
Today at 8:15 AM
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J
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116 problems
momed66   18
N Aug 31, 2020 by Gaussian_cyber
Source: By Mohammad Jafari
Problem 85 was one of IRAN TST 2011 problems.
18 replies
momed66
May 14, 2011
Gaussian_cyber
Aug 31, 2020
J
116 problems
G H J
Reveal topic tags
function
inequalities
induction
Problem Sets
L
G H BBookmark kLocked kLocked NReply
Source: By Mohammad Jafari
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momed66
14 posts
momed66
#1 h  May 14, 2011, 4:40 PM • 23 Y
Y by NewAlbionAcademy, airplanes1, Neversmile123, bryanbr2, Yousefi, sanabhai, HMoghaddas, dantx5, Jerry37284, Wizard_32, fjm30, Functional_equation, Math-wiz, Adventure10, Mango247, and 8 other users
Problem 85 was one of IRAN TST 2011 problems.
Attachments:
116 problems.pdf (215kb)
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  May 14, 2011, 4:42 PM • 2 Y
Y by Adventure10, Mango247
hello, nice problems, thank you.
Sonnhard.
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goodar2006
1347 posts
goodar2006
#3 h  May 14, 2011, 5:51 PM • 2 Y
Y by Adventure10, Mango247
Thanks Mr.Jafari for the problems.
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Amir Hossein
5452 posts
Amir Hossein
#4 h  May 14, 2011, 6:03 PM • 2 Y
Y by Adventure10, Mango247
Great collection, thanks Mr. Jafari! :)
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Ligouras
826 posts
Ligouras
#5 h  May 14, 2011, 8:42 PM • 2 Y
Y by Adventure10, Mango247
Nice collection, thanks Mr. Jafari!!! :)
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abch42
127 posts
abch42
#6 h  May 15, 2011, 1:57 PM • 2 Y
Y by Adventure10, Mango247
Very nice! Thanks Mr.Jafari :)
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xiyuguzu
25 posts
xiyuguzu
#7 h  May 20, 2011, 1:02 AM • 2 Y
Y by Adventure10, Mango247
where to find the solutions?
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Merlinaeus
162 posts
Merlinaeus
#8 h  May 21, 2011, 5:08 PM • 1 Y
Y by Adventure10
Mods - What's your preferred way for us to discuss these problems?

Dang it, I can't do #1 !

Merlin
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applepi2000
2226 posts
applepi2000
#9 h  Nov 8, 2011, 10:26 PM • 6 Y
Y by Yoni2b, socrates, Amir Hossein, Adventure10, Mango247, and 1 other user
Here are a few of my solutions :lol:
Problem 2
Problem 11
Problem 9
Problem 37
Problem 39
Problem 69
Problem 70
Problem 71
Problem 79
Problem 105
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socrates
2105 posts
socrates
#10 h  Nov 9, 2011, 3:23 PM • 3 Y
Y by Amir Hossein, Adventure10, and 1 other user
Problem 43
Problem 30
Problem 31
Problem 26
Problem 81, Stronger Version
Problem 60
Problem 85
Problem 42
Problem 6
Problem 5
Problem 2
Problem 20
Problem 4
Problem 34
Problem 33

Other problems:
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378717
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378367
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378164
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=378718
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socrates
2105 posts
socrates
#11 h  Nov 9, 2011, 8:12 PM • 5 Y
Y by Amir Hossein, applepi2000, kienthuckiquai, Adventure10, Mango247
Problem 7
Problem 8
Problem 15
Problem 16
Problem 17
Problem 19
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 27
Problem 29
Problem 12
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applepi2000
2226 posts
applepi2000
#12 h  Nov 10, 2011, 12:19 AM • 4 Y
Y by socrates, Amir Hossein, Adventure10, Mango247
Problem 49
Problem 50
Problem 52
Problem 59
Problem 62
Problem 64
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socrates
2105 posts
socrates
#13 h  Nov 10, 2011, 2:19 AM • 5 Y
Y by applepi2000, Amir Hossein, a123, Adventure10, Mango247
Problem 18
Problem 28
Problem 36
Problem 38
Problem 44
Problem 45
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Amir Hossein
5452 posts
Amir Hossein
#14 h  Nov 10, 2011, 9:14 AM • 1 Y
Y by Adventure10
Please post your solutions in this topic.
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Functional_equation
530 posts
Functional_equation
#15 h  Aug 31, 2020, 1:39 PM • 1 Y
Y by Mango247
Great handout! :)
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Functional_equation
530 posts
Functional_equation
#17 h  Aug 31, 2020, 1:47 PM • 4 Y
Y by Bumblebee60, amar_04, Wizard0001, Mango247
In fact, I didn't do it to congratulate him. I just wanted other people (newcomers to Aops) to see it :)
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franzliszt
23531 posts
franzliszt
#18 h  Aug 31, 2020, 1:54 PM • 1 Y
Y by Mango247
Ya good bump man! :)
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SardorNarpay
106 posts
SardorNarpay
#19 h  Aug 31, 2020, 3:16 PM
Y by
Thank you very much
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Gaussian_cyber
162 posts
Gaussian_cyber
#20 h  Aug 31, 2020, 3:21 PM • 2 Y
Y by Mango247, Mango247
You can see This. It's is the newer version Ig.
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