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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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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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Marking vertices in splitted triangle
mathisreal   2
N an hour 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
an hour ago
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distance of a point from incircle equals to a diameter of incircle
parmenides51   5
N an hour 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
an hour 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 5 hours ago 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
1 viewing
parmenides51
Jul 4, 2019
TiagoCavalcante
5 hours ago
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Functional Inequality Implies Uniform Sign
peace09   33
N 5 hours ago by ezpotd
Source: 2023 ISL A2
Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\]for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$.

Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.
33 replies
peace09
Jul 17, 2024
ezpotd
5 hours ago
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Trapezium problem very nice
manlio   2
N 6 hours 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
Yesterday at 11:00 AM
alexheinis
6 hours ago
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Labelling edges of Kn
oVlad   1
N 6 hours ago by TopGbulliedU
Source: Romania Junior TST 2025 Day 2 P3
Let $n\geqslant 3$ be an integer. Ion draws a regular $n$-gon and all its diagonals. On every diagonal and edge, Ion writes a positive integer, such that for any triangle formed with the vertices of the $n$-gon, one of the numbers on its edges is the sum of the two other numbers on its edges. Determine the smallest possible number of distinct values that Ion can write.
1 reply
oVlad
May 6, 2025
TopGbulliedU
6 hours ago
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c^a + a = 2^b
Havu   8
N 6 hours ago by MathematicalArceus
Find $a, b, c\in\mathbb{Z}^+$ such that $a,b,c$ coprime, $a + b = 2c$ and $c^a + a = 2^b$.
8 replies
Havu
May 10, 2025
MathematicalArceus
6 hours ago
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Concurrence of lines defined by intersections of circles
Lukaluce   1
N Yesterday at 8:28 PM 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
Yesterday at 8:28 PM
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Factorial Divisibility
Aryan-23   45
N Yesterday at 8:28 PM 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
Yesterday at 8:28 PM
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Multiple of multinomial coefficient is an integer
orl   14
N Yesterday at 8:23 PM 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
Yesterday at 8:23 PM
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Functional Equation from IMO
prtoi   1
N Yesterday at 8:22 PM 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
Yesterday at 8:07 PM
KAME06
Yesterday at 8:22 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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What are vectors?
SomeonecoolLovesMaths   3
N Apr 20, 2025 by programjames1
How does one define vectors with mathematical rigority? Most of the time it is stated as an object with a magnitude and a direction, but it in itself is very vague in my opinion. Like how do you define direction and magnitude itself?
3 replies
SomeonecoolLovesMaths
Apr 20, 2025
programjames1
Apr 20, 2025
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What are vectors?
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Reveal topic tags
vector
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SomeonecoolLovesMaths
3263 posts
SomeonecoolLovesMaths
#1 h  Apr 20, 2025, 4:19 AM
Y by
How does one define vectors with mathematical rigority? Most of the time it is stated as an object with a magnitude and a direction, but it in itself is very vague in my opinion. Like how do you define direction and magnitude itself?
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greenturtle3141
3559 posts
greenturtle3141
#2 h  Apr 20, 2025, 5:45 AM
Y by
For the sort of "vectors" you are probably thinking of, they are simply elements of $\mathbb{R}^n$.

For example, if the net force on an object is "10 units of force directed upward", that would mathematically be something like $(0,0,10)$.

(More generally and abstractly, a mathematician would define a vector as an element of a vector space, but that's likely out of the scope of this discussion.)
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SomeonecoolLovesMaths
3263 posts
SomeonecoolLovesMaths
#3 h  Apr 20, 2025, 5:53 AM
Y by
greenturtle3141 wrote:
For the sort of "vectors" you are probably thinking of, they are simply elements of $\mathbb{R}^n$.

For example, if the net force on an object is "10 units of force directed upward", that would mathematically be something like $(0,0,10)$.

(More generally and abstractly, a mathematician would define a vector as an element of a vector space, but that's likely out of the scope of this discussion.)

Okay I get it. It is probably out of my league to grasp the concept of vectors but I just wanted to get an idea of what they truly are. Thanks! :D
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programjames1
3046 posts
programjames1
#4 h  Apr 20, 2025, 7:24 AM
Y by
See this discussion from a few months ago:

https://artofproblemsolving.com/community/c4h3520259p34184343
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