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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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[*]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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Trapezium problem very nice
manlio   2
N an hour ago 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
Today at 11:00 AM
alexheinis
an hour ago
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Concurrence of lines defined by intersections of circles
Lukaluce   1
N 2 hours ago by sarjinius
Source: 2025 Macedonian Balkan Math Olympiad TST Problem 2
Let $\triangle ABC$ be an acute-angled triangle and $A_1, B_1$, and $C_1$ be the feet of the altitudes from $A, B$, and $C$, respectively. On the rays $AA_1, BB_1$, and $CC_1$, we have points $A_2, B_2$, and $C_2$ respectively, lying outside of $\triangle ABC$, such that
\[\frac{A_1A_2}{AA_1} = \frac{B_1B_2}{BB_1} = \frac{C_1C_2}{CC_1}.\]If the intersections of $B_1C_2$ and $B_2C_1$, $C_1A_2$ and $C_2A_1$, and $A_1B_2$ and $A_2B_1$ are $A', B'$, and $C'$ respectively, prove that $AA', BB'$, and $CC'$ have a common point.
1 reply
Lukaluce
Apr 14, 2025
sarjinius
2 hours ago
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Factorial Divisibility
Aryan-23   45
N 2 hours ago by MathematicalArceus
Source: IMO SL 2022 N2
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
45 replies
Aryan-23
Jul 9, 2023
MathematicalArceus
2 hours ago
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Multiple of multinomial coefficient is an integer
orl   14
N 2 hours ago by mickeymouse7133
Source: Romanian Master in Mathematics 2009, Problem 1
For $ a_i \in \mathbb{Z}^ +$, $ i = 1, \ldots, k$, and $ n = \sum^k_{i = 1} a_i$, let $ d = \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i = 1} (a_i!)}$ is an integer.

Dan Schwarz, Romania
14 replies
orl
Mar 7, 2009
mickeymouse7133
2 hours ago
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Functional Equation from IMO
prtoi   1
N 2 hours ago by KAME06
Source: IMO
Question: $f(2a)+2f(b)=f(f(a+b))$
Solve for f:Z-->Z
My solution:
At a=0, $f(0)+2f(b)=f(f(b))$
Take t=f(b) to get $f(0)+2t=f(t)$
Therefore, f(x)=2x+n where n=f(0)
Could someone please clarify if this is right or wrong?
1 reply
prtoi
2 hours ago
KAME06
2 hours ago
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can you solve this..?
Jackson0423   1
N 2 hours ago by GreekIdiot
Source: Own

Find the number of integer pairs \( (x, y) \) satisfying the equation
\[ 4x^2 - 3y^2 = 1 \]such that \( |x| \leq 2025 \).
1 reply
Jackson0423
May 8, 2025
GreekIdiot
2 hours ago
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Gergonne point Harmonic quadrilateral
niwobin   0
2 hours ago
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)
0 replies
niwobin
2 hours ago
0 replies
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Concurrent in a pyramid
vanstraelen   1
N 2 hours ago 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
2 hours ago
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bisector of <BAC _|_AD, trapezium, AB = BE, AC = DE NZMO 2021 R1 p2
parmenides51   3
N 2 hours ago 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
2 hours ago
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Pertenacious Polynomial Problem
BadAtCompetitionMath21420   4
N 3 hours ago 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
Today at 3:13 AM
soryn
3 hours ago
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Combi that will make you question every choice in your life so far
blug   1
N 3 hours ago by HotSinglesInYourArea
$A$ and $B$ are standing in front of the room in which there is $C$. They know that there is a chessboard in the room and that on every square there is a coin. Every coin is black on one side and white on the other side and is flipped randomly. $A$ enters the room and then $C$ points at exactly one square on the chessboard. After that, $A$ must flip exactly one coin of his choice on the chessboard to the other side and leave. Finally, $B$ enters the room ($A$ and $B$ haven't met again after $A$ entered the room) and he has to guess which square did $C$ point at.
What strategy do $A$ and $B$ have that will make this happen every time?
1 reply
blug
5 hours ago
HotSinglesInYourArea
3 hours ago
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Functional equation
Pmshw   17
N 3 hours ago by arzhang2001
Source: Iran 2nd round 2022 P2
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any real value of $x,y$ we have:
$$f(xf(y)+f(x)+y)=xy+f(x)+f(y)$$
17 replies
Pmshw
May 8, 2022
arzhang2001
3 hours ago
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Hard Geometry
Jalil_Huseynov   3
N 3 hours ago by bin_sherlo
Source: DGO 2021, Individual stage, Day1 P3
Let triangle $ABC$ be a triangle with incenter $I$ and circumcircle $\Omega$ with circumcenter $O$. The incircle touches $CA, AB$ at $E, F$ respectively. $R$ is another intersection point of external bisector of $\angle BAC$ with $\Omega$, and $T$ is $\text{A-mixtillinear}$ incircle touch point to $\Omega$. Let $W, X, Z$ be points lie on $\Omega$. $RX$ intersect $AI$ at $Y$ . Assume that $R \ne X$. Suppose that $E, F, X, Y$ and $W, Z, E, F$ are concyclic, and $AZ, EF, RX$ are concurrent.
Prove that
$\bullet$ $AZ, RW, OI$ are concurrent.
$\bullet$ $\text{A-symmedian}$, tangent line to $\Omega$ at $T$ and $WZ$ are concurrent.

Proporsed by wassupevery1 and k12byda5h
3 replies
Jalil_Huseynov
Dec 26, 2021
bin_sherlo
3 hours ago
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Algebra form IMO Shortlist
Abbas11235   35
N 3 hours ago by ezpotd
Source: IMO Shortlist 2017 A2
Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[list]
[*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
[*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are
two (not necessarily distinct) numbers from the first line.
[*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
[/list]
Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
35 replies
Abbas11235
Jul 10, 2018
ezpotd
3 hours ago
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Geometry question
Dheckob   2
N Apr 13, 2017 by Immanuel Bonfils
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.
2 replies
Dheckob
Apr 12, 2017
Immanuel Bonfils
Apr 13, 2017
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Geometry question
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Dheckob
7 posts
Dheckob
#1 h  Apr 12, 2017, 11:30 PM • 1 Y
Y by Adventure10
$ABCD$ is a rectangle. $E$ is a point on $AB$ between $A$ and $B$, and $F$ is a point on $AD$ between $A$ and $D$. The area of the triangle $EBC$ is $16$, the area of the triangle $EAF$ is $12$ and the area of the triangle $FDC$ is 30. Find the area of the triangle $EFC$.
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mathman3880
423 posts
mathman3880
#2 h  Apr 13, 2017, 12:43 AM • 2 Y
Y by Adventure10, Mango247
sol
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Immanuel Bonfils
2094 posts
Immanuel Bonfils
#3 h  Apr 13, 2017, 2:55 AM • 2 Y
Y by Adventure10, Mango247
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This post has been edited 2 times. Last edited by Immanuel Bonfils, Apr 13, 2017, 3:02 AM
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