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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
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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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Problem 1, BMO 2020
dangerousliri   40
N 44 minutes ago by Giant_PT
Source: Problem 1, BMO 2020
Let $ABC$ be an acute triangle with $AB=AC$, let $D$ be the midpoint of the side $AC$, and let $\gamma$ be the circumcircle of the triangle $ABD$. The tangent of $\gamma$ at $A$ crosses the line $BC$ at $E$. Let $O$ be the circumcenter of the triangle $ABE$. Prove that midpoint of the segment $AO$ lies on $\gamma$.

Proposed by Sam Bealing, United Kingdom
40 replies
dangerousliri
Nov 1, 2020
Giant_PT
44 minutes ago
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Terrifying "2018 \times 2019" board
IndoMathXdZ   20
N an hour ago by HamstPan38825
Source: APMO 2019 P4
Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average.
Is it always possible to make the numbers in all squares become the same after finitely many turns?
20 replies
IndoMathXdZ
Jun 11, 2019
HamstPan38825
an hour ago
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inequality
mathematical-forest   3
N 2 hours ago by RainbowNeos
For positive real intengers $x_{1} ,x_{2} ,\cdots,x_{n} $, such that $\prod_{i=1}^{n} x_{i} =1$
proof:
$$\sum_{i=1}^{n} \frac{1}{1+\sum _{j\ne i}x_{j} } \le 1$$
3 replies
mathematical-forest
May 15, 2025
RainbowNeos
2 hours ago
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At least k points of S equidistant from P
orl   9
N 2 hours ago by Twan
Source: IMO 1989/3 , ISL 20, ILL 66
Let $ n$ and $ k$ be positive integers and let $ S$ be a set of $ n$ points in the plane such that

i.) no three points of $ S$ are collinear, and

ii.) for every point $ P$ of $ S$ there are at least $ k$ points of $ S$ equidistant from $ P.$

Prove that:
\[ k < \frac {1}{2} + \sqrt {2 \cdot n} \]
9 replies
orl
Nov 19, 2005
Twan
2 hours ago
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Gergonne point Harmonic quadrilateral
niwobin   1
N 2 hours ago by on_gale
Triangle ABC has incircle touching the sides at D, E, F as shown.
AD, BE, CF concurrent at Gergonne point G.
BG and CG cuts the incircle at X and Y, respectively.
AG cuts the incircle at K.
Prove: K, X, D, Y form a harmonic quadrilateral. (KX/KY = DX/DY)
1 reply
niwobin
Yesterday at 8:17 PM
on_gale
2 hours ago
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Find the minimum
sqing   8
N 3 hours ago by sqing
Source: China Shandong High School Mathematics Competition 2025 Q4
Let $ a,b,c>0,abc>1$. Find the minimum value of $ \frac {abc(a+b+c+8)}{abc-1}. $
8 replies
sqing
Yesterday at 9:12 AM
sqing
3 hours ago
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Interesting inequalities
sqing   3
N 3 hours ago by sqing
Source: Own
Let $ a,b >0 $ and $ a^2-ab+b^2\leq 1 $ . Prove that
$$a^4 +b^4+\frac{a }{b +1}+ \frac{b }{a +1} \leq 3$$$$a^3 +b^3+\frac{a^2}{b^2+1}+ \frac{b^2}{a^2+1} \leq 3$$$$a^4 +b^4-\frac{a}{b+1}-\frac{b}{a+1} \leq 1$$$$a^4+b^4 -\frac{a^2}{b^2+1}- \frac{b^2}{a^2+1}\leq 1$$$$a^3+b^3 -\frac{a^3}{b^3+1}- \frac{b^3}{a^3+1}\leq 1$$
3 replies
sqing
May 9, 2025
sqing
3 hours ago
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Marking vertices in splitted triangle
mathisreal   2
N 3 hours ago by sopaconk
Source: Mexico
Let $n$ be a positive integer. Consider a figure of a equilateral triangle of side $n$ and splitted in $n^2$ small equilateral triangles of side $1$. One will mark some of the $1+2+\dots+(n+1)$ vertices of the small triangles, such that for every integer $k\geq 1$, there is not any trapezoid(trapezium), whose the sides are $(1,k,1,k+1)$, with all the vertices marked. Furthermore, there are no small triangle(side $1$) have your three vertices marked. Determine the greatest quantity of marked vertices.
2 replies
mathisreal
Feb 7, 2022
sopaconk
3 hours ago
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distance of a point from incircle equals to a diameter of incircle
parmenides51   5
N 3 hours ago by Captainscrubz
Source: 2019 Oral Moscow Geometry Olympiad grades 8-9 p1
In the triangle $ABC, I$ is the center of the inscribed circle, point $M$ lies on the side of $BC$, with $\angle BIM = 90^o$. Prove that the distance from point $M$ to line $AB$ is equal to the diameter of the circle inscribed in triangle $ABC$
5 replies
parmenides51
May 21, 2019
Captainscrubz
3 hours ago
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f(a + b) = f(a) + f(b) + f(c) + f(d) in N-{O}, with 2ab = c^2 + d^2
parmenides51   8
N Yesterday at 9:55 PM by TiagoCavalcante
Source: RMM Shortlist 2016 A1
Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.
8 replies
parmenides51
Jul 4, 2019
TiagoCavalcante
Yesterday at 9:55 PM
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Trapezium problem very nice
manlio   2
N Yesterday at 8:52 PM by alexheinis
Given trapezium ABCD with basis AB and CD parallel. Choose a point E on side BC and a point F on side AD such that AE Is parallel to FC . Prove that DE Is parallel to FB.
2 replies
manlio
Yesterday at 11:00 AM
alexheinis
Yesterday at 8:52 PM
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Concurrent in a pyramid
vanstraelen   1
N Yesterday at 7:53 PM by vanstraelen

Given a pyramid $(T,ABCD)$ where $ABCD$ is a parallelogram.
The intersection of the diagonals of the base is point $S$.
Point $A$ is connected to the midpoint of $[CT]$, point $B$ to the midpoint of $[DT]$,
point $C$ to the midpoint of $[AT]$ and point $D$ to the midpoint of $[BT]$.
a) Prove: the four lines are concurrent in a point $P$.
b) Calulate $\frac{TS}{TP}$.
1 reply
vanstraelen
May 10, 2025
vanstraelen
Yesterday at 7:53 PM
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bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2
parmenides51   3
N Yesterday at 7:49 PM by LeYohan
Let $ABCD$ be a trapezium such that $AB\parallel CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB = BE$ and $AC = DE$. Prove that the internal angle bisector of $\angle BAC$ is perpendicular to $AD$.
3 replies
parmenides51
Sep 20, 2021
LeYohan
Yesterday at 7:49 PM
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Pertenacious Polynomial Problem
BadAtCompetitionMath21420   4
N Yesterday at 7:40 PM by soryn
Let the polynomial $P(x) = x^3-x^2+px-q$ have real roots and real coefficients with $q>0$. What is the maximum value of $p+q$?

This is a problem I made for my math competition, and I wanted to see if someone would double-check my work (No Mike allowed):

solution
Is this solution good?
4 replies
BadAtCompetitionMath21420
Yesterday at 3:13 AM
soryn
Yesterday at 7:40 PM
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Geometry books
T.Mousavidin   4
N Apr 30, 2025 by compoly2010
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,
4 replies
T.Mousavidin
Apr 29, 2025
compoly2010
Apr 30, 2025
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Geometry books
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Reveal topic tags
projective geometry
geometry
3D geometry
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T.Mousavidin
3 posts
T.Mousavidin
#1 h  Apr 29, 2025, 4:25 PM • 1 Y
Y by M.Roueintan
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,
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martianrunner
207 posts
martianrunner
#2 h  Apr 29, 2025, 6:26 PM
Y by
egmo is always a good pick if you are at that level
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VivaanKam
167 posts
VivaanKam
#3 h  Apr 29, 2025, 7:46 PM
Y by
I don't know If it contains those but the aops book Introduction To Geometry by Richard Rusczyk is pretty good.
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Shan3t
380 posts
Shan3t
#4 h  Apr 29, 2025, 11:06 PM
Y by
T.Mousavidin wrote:
Hello, I wanted to ask if anybody knows some good books for geometry that has these topics in:
Desargues's Theorem, Projective geometry, 3D geometry,

EGMO(Euclidean Geometry by Evan Chen) is a very good book, and i believe it has all of those topics :D(3D geo, and projective for sure, idk abt Desargue's Theorem tho...)
This post has been edited 1 time. Last edited by Shan3t, Apr 29, 2025, 11:07 PM
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compoly2010
5 posts
compoly2010
#5 h  Apr 30, 2025, 12:10 AM
Y by
Another alternative is Geometry Revisited (Coxeter), which in my opinion holds a terser take to your desires. In particular, it also covers stereographic and gnomonic projections.

Personally, however, I find EGMO covers your topics sufficiently for olympiad purposes.

Both, if I recall correctly, do not cover some more advanced techniques such as MMP, or LinPoP.
This post has been edited 1 time. Last edited by compoly2010, Apr 30, 2025, 1:06 AM
Reason: clarity
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