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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 2, 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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Vasc = 1?
Li4   7
N 6 minutes ago by internationalnick123456
Source: 2025 Taiwan TST Round 3 Independent Study 1-N
Find all integer tuples $(a, b, c)$ such that
\[(a^2 + b^2 + c^2)^2 = 3(a^3b + b^3c + c^3a) + 1. \]
Proposed by Li4, Untro368, usjl and YaWNeeT.
7 replies
Li4
Apr 26, 2025
internationalnick123456
6 minutes ago
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NT Functional Equation
mkultra42   1
N 10 minutes ago by mkultra42
Find all strictly increasing functions \(f: \mathbb{N} \to \mathbb{N}\) satsfying \(f(1)=1\) and:

\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]
1 reply
mkultra42
Apr 25, 2025
mkultra42
10 minutes ago
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Combination and sum
Kunihiko_Chikaya   1
N 11 minutes ago by Mathzeus1024
Source: The Jikei University School of Medicine entrance exam 2006
Let $n$ be positive integer.Prove the following equality.

\[ \sum_{r=1}^n (-1)^r\ _n C_r\ \frac{1}{r}=-\sum_{r=1}^n \frac{1}{r} \]
1 reply
Kunihiko_Chikaya
Jan 29, 2006
Mathzeus1024
11 minutes ago
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QM + HM >= AM + GM?
greenturtle3141   5
N 13 minutes ago by bansongbada
Prove or disprove:
$$QM+HM \geq AM + GM$$Where $QM$ is quadratic mean $\sqrt{\dfrac{a_1^2+a_2^2+\cdots+a_n^2}{n}}$, $HM$ is harmonic mean $\dfrac{n}{\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_n}}$, $AM$ is arithmetic mean $\dfrac{a_1+a_2+\cdots+a_n}{n}$, and $GM$ is geometric mean $\sqrt[n]{a_1a_2\cdots a_n}$.
5 replies
greenturtle3141
May 16, 2019
bansongbada
13 minutes ago
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Coincide
giangtruong13   1
N 14 minutes ago by giangtruong13
Source: Hanoi Specialized School's Math Test (Round 2 - Phase 1)
Let $ABCD$ be a trapezoid inscribed in circle $(O)$, $AD||BC, AD < BC$. Let $P$ is the symmetric point of $A$ across $BC$, $AP$ intersects $BC$ at $K$. Let $M$ is midpoint of $BC$ and $H$ is orthocenter of triangle $ABC$. On $BD$ take a point $F$ so that $AF||HM$. Prove that: $ FK,AC,PD$ coincide
1 reply
giangtruong13
Yesterday at 4:05 PM
giangtruong13
14 minutes ago
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About my new website
Samujjal101   0
24 minutes ago
Hi everybody!
I'm registering some of the finest minds in math into my website.. it's not completely developed.. but still if you want we would be very grateful to have you!
Text to display
Maths-matchmaker is a website for connecting math minds together with a mission to unite together. It uses a matching algorithm to match 1:1 with like minded peers based on their interests or topics in math
0 replies
Samujjal101
24 minutes ago
0 replies
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JBMO Shortlist 2021 G4
Lukaluce   3
N 29 minutes ago by S11_DooronbekovNurakhmed
Source: JBMO Shortlist 2021
Let $ABCD$ be a convex quadrilateral with $\angle B = \angle D = 90^{\circ}$. Let $E$ be the point of intersection of $BC$ with $AD$ and let $M$ be the midpoint of $AE$. On the extension of $CD$, beyond the point $D$, we pick a point $Z$ such that $MZ = \frac{AE}{2}$. Let $U$ and $V$ be the projections of $A$ and $E$ respectively on $BZ$. The circumcircle of the triangle $DUV$ meets again $AE$ at the point $L$. If $I$ is the point of intersection of $BZ$ with $AE$, prove that the lines $BL$ and $CI$ intersect on the line $AZ$.
3 replies
Lukaluce
Jul 2, 2022
S11_DooronbekovNurakhmed
29 minutes ago
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Determine all the sets of six consecutive positive integers
sqing   7
N 40 minutes ago by Adventure1000
Source: JBMO 2017, Q1
Determine all the sets of six consecutive positive integers such that the product of some two of them . added to the product of some other two of them is equal to the product of the remaining two numbers.
7 replies
sqing
Jun 26, 2017
Adventure1000
40 minutes ago
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Polygon with minimum internal angle 120^\circ
Kunihiko_Chikaya   1
N 43 minutes ago by Mathzeus1024
How many sides can have the polygon with minimum internal angle of $ 120^\circ$ by adding every $ 5^\circ$?
Sorry for my poor English.
1 reply
Kunihiko_Chikaya
Aug 9, 2007
Mathzeus1024
43 minutes ago
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One more problem defined only with lines
Assassino9931   2
N an hour ago by sami1618
Source: Balkan MO 2024 Shortlist G6
Let $ABC$ be a triangle and the points $K$ and $L$ on $AB$, $M$ and $N$ on $BC$, and $P$ and $Q$ on $AC$ be such that $AK = LB < \frac{1}{2}AB, BM = NC < \frac{1}{2}BC$ and $CP = QA < \frac{1}{2}AC$. The intersections of $KN$ with $MQ$ and $LP$ are $R$ and $T$ respectively, and the intersections of $NP$ with $LM$ and $KQ$ are $D$ and $E$, respectively. Prove that the lines $DR, BE$ and $CT$ are concurrent.
2 replies
1 viewing
Assassino9931
Yesterday at 10:31 PM
sami1618
an hour ago
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Two parallels
jayme   3
N an hour ago by ND_
Source: own?
Dear Mathlinkers,

1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. X the point of intersection of the parallel to AB through Q and BC
7. Y the point of intersection of the segment AX and (A)

Prove : CY is parallel to IP.

Jean-Louis
3 replies
jayme
Today at 8:25 AM
ND_
an hour ago
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Interesting inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b,c>0 . $ Prove that
$$\frac{a}{b}+ \frac{kb^2}{c^2} + \frac{c}{a}\geq 5\sqrt[5]{\frac{k}{16}}$$Where $ k >0. $
$$\frac{a}{b}+ \frac{16b^2}{c^2} + \frac{c}{a}\geq 5$$$$\frac{a}{b}+ \frac{ b^2}{2c^2} + \frac{c}{a}\geq \frac{5}{2} $$
1 reply
sqing
an hour ago
sqing
an hour ago
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Junior Balkan Mathematical Olympiad 2021- P3
Lukaluce   35
N an hour ago by Rayvhs
Source: JBMO 2021
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$.
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.
35 replies
Lukaluce
Jul 1, 2021
Rayvhs
an hour ago
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amazing balkan combi
egxa   3
N an hour ago by CatinoBarbaraCombinatoric
Source: BMO 2025 P4
There are $n$ cities in a country, where $n \geq 100$ is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities $A$ and $B$ we define:

$(i)$ A $\emph{path}$ between $A$ and $B$ as a sequence of distinct cities $A = C_0, C_1, \dots, C_k, C_{k+1} = B$, $k \geq 0$, such that there are direct flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$;
$(ii)$ A $\emph{long path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has more cities;
$(iii)$ A $\emph{short path}$ between $A$ and $B$ as a path between $A$ and $B$ such that no other path between $A$ and $B$ has fewer cities.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). Let $F$ be the total number of pairs of cities in the country that are connected by direct flights. In terms of $n$, find all possible values $F$

Proposed by David-Andrei Anghel, Romania.
3 replies
+1 w
egxa
Yesterday at 1:57 PM
CatinoBarbaraCombinatoric
an hour ago
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acute triangle
syncmaster   1
N Jan 8, 2013 by dan23
Source: ABC
in acute triangle ABC;D,E,F are foots of altitudes to BC,AC,AB respectively. H-ortocenter.
Prove that (AH/AD)+(BH/BE)+(CH/CF)=2
1 reply
syncmaster
Jan 8, 2013
dan23
Jan 8, 2013
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acute triangle
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Reveal topic tags
geometry unsolved
geometry
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Source: ABC
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syncmaster
14 posts
syncmaster
#1 h  Jan 8, 2013, 6:57 PM • 2 Y
Y by Adventure10, Mango247
in acute triangle ABC;D,E,F are foots of altitudes to BC,AC,AB respectively. H-ortocenter.
Prove that (AH/AD)+(BH/BE)+(CH/CF)=2
Z K Y
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dan23
32 posts
dan23
#2 h  Jan 8, 2013, 7:48 PM • 2 Y
Y by Adventure10, Mango247
obviously we have : $S_{\bigtriangleup BCH}+S_ {\bigtriangleup ACH} +S_{\bigtriangleup ABH}=S_ {\bigtriangleup ABC}$ (1)

and we have three equations like this for each of$ \bigtriangleup BCH \& \bigtriangleup ACH \& \bigtriangleup ABH:$

$\frac{AH}{AD}=\frac{(AD-DH)}{AD}=\frac{(S_{\bigtriangleup ABC} - S_{\bigtriangleup BCH})}{S_{\bigtriangleup ABC}} =1- \frac{S _{\bigtriangleup BCH}}{S_{\bigtriangleup ABC}}$

Add the R.H.S. of these 3 together and 3 L.H.S. together and use (1)
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