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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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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
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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Divisibility on 101 integers
BR1F1SZ   4
N 9 minutes ago by BR1F1SZ
Source: Argentina Cono Sur TST 2024 P2
There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.
4 replies
BR1F1SZ
Aug 9, 2024
BR1F1SZ
9 minutes ago
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2^x+3^x = yx^2
truongphatt2668   2
N 19 minutes ago by CM1910
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
2 replies
truongphatt2668
Yesterday at 3:38 PM
CM1910
19 minutes ago
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Prove perpendicular
shobber   29
N 31 minutes ago by zuat.e
Source: APMO 2000
Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced.

Prove that $QO$ is perpendicular to $BC$.
29 replies
shobber
Apr 1, 2006
zuat.e
31 minutes ago
J
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The smallest of sum of elements
hlminh   1
N an hour ago by nguyenhuybao_06
Let $S=\{1,2,...,2014\}$ and $X\subset S$ such that for all $a,b\in X,$ if $a+b\leq 2014$ then $a+b\in X.$ Find the smallest of sum of all elements of $X.$
1 reply
hlminh
an hour ago
nguyenhuybao_06
an hour ago
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Number theory or function ?
matematikator   15
N 2 hours ago by YaoAOPS
Source: IMO ShortList 2004, algebra problem 3
Does there exist a function $s\colon \mathbb{Q} \rightarrow \{-1,1\}$ such that if $x$ and $y$ are distinct rational numbers satisfying ${xy=1}$ or ${x+y\in \{0,1\}}$, then ${s(x)s(y)=-1}$? Justify your answer.

Proposed by Dan Brown, Canada
15 replies
matematikator
Mar 18, 2005
YaoAOPS
2 hours ago
J
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Bounding number of solutions for floor function equation
Ciobi_   1
N 2 hours ago by sarjinius
Source: Romania NMO 2025 9.3
Let $n \geq 2$ be a positive integer. Consider the following equation: \[ \{x\}+\{2x\}+ \dots + \{nx\} = \lfloor x \rfloor + \lfloor 2x \rfloor + \dots + \lfloor 2nx \rfloor\]a) For $n=2$, solve the given equation in $\mathbb{R}$.
b) Prove that, for any $n \geq 2$, the equation has at most $2$ real solutions.
1 reply
Ciobi_
Apr 2, 2025
sarjinius
2 hours ago
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Inequalities
idomybest   3
N 4 hours ago by damyan
Source: The Interesting Around Technical Analysis Three Variable Inequalities
The problem is in the attachment below.
3 replies
idomybest
Oct 15, 2021
damyan
4 hours ago
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A game optimization on a graph
Assassino9931   2
N 5 hours ago by dgrozev
Source: Bulgaria National Olympiad 2025, Day 2, Problem 6
Let \( X_0, X_1, \dots, X_{n-1} \) be \( n \geq 2 \) given points in the plane, and let \( r > 0 \) be a real number. Alice and Bob play the following game. Firstly, Alice constructs a connected graph with vertices at the points \( X_0, X_1, \dots, X_{n-1} \), i.e., she connects some of the points with edges so that from any point you can reach any other point by moving along the edges.Then, Alice assigns to each vertex \( X_i \) a non-negative real number \( r_i \), for \( i = 0, 1, \dots, n-1 \), such that $\sum_{i=0}^{n-1} r_i = 1$. Bob then selects a sequence of distinct vertices \( X_{i_0} = X_0, X_{i_1}, \dots, X_{i_k} \) such that \( X_{i_j} \) and \( X_{i_{j+1}} \) are connected by an edge for every \( j = 0, 1, \dots, k-1 \). (Note that the length $k \geq 0$ is not fixed and the first selected vertex always has to be $X_0$.) Bob wins if
\[ \frac{1}{k+1} \sum_{j=0}^{k} r_{i_j} \geq r; \]otherwise, Alice wins. Depending on \( n \), determine the largest possible value of \( r \) for which Bob has a winning strategy.
2 replies
Assassino9931
Apr 8, 2025
dgrozev
5 hours ago
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Composite sum
rohitsingh0812   39
N 5 hours ago by YaoAOPS
Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
39 replies
rohitsingh0812
Jun 3, 2006
YaoAOPS
5 hours ago
J
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Factor of P(x)
Brut3Forc3   20
N 6 hours ago by IceyCold
Source: 1976 USAMO Problem 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\] prove that $ x-1$ is a factor of $ P(x)$.
20 replies
Brut3Forc3
Apr 4, 2010
IceyCold
6 hours ago
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The old one is gone.
EeEeRUT   9
N Today at 1:26 PM by Jupiterballs
Source: EGMO 2025 P2
An infinite increasing sequence $a_1 < a_2 < a_3 < \cdots$ of positive integers is called central if for every positive integer $n$ , the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1, b_2, b_3, \dots$ of positive integers such that for every central sequence $a_1, a_2, a_3, \dots, $ there are infinitely many positive integers $n$ with $a_n = b_n$.
9 replies
EeEeRUT
Apr 16, 2025
Jupiterballs
Today at 1:26 PM
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Complicated FE
XAN4   0
Today at 11:53 AM
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
0 replies
XAN4
Today at 11:53 AM
0 replies
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interesting function equation (fe) in IR
skellyrah   1
N Today at 11:37 AM by CrazyInMath
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
1 reply
skellyrah
Today at 9:51 AM
CrazyInMath
Today at 11:37 AM
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Find maximum area of right triangle
jl_   1
N Today at 11:18 AM by navier3072
Source: Malaysia IMONST 2 2023 (Primary) P4
Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.
1 reply
jl_
Today at 10:33 AM
navier3072
Today at 11:18 AM
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<DAC =<ABE iff <AFC = <BDA , midpoints of sides of triangle related
parmenides51   1
N May 25, 2020 by badumtsss
Source: 2010 Romania JBMO TST5 P2
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$
1 reply
parmenides51
May 25, 2020
badumtsss
May 25, 2020
J
<DAC =<ABE iff <AFC = <BDA , midpoints of sides of triangle related
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Reveal topic tags
midpoints
geometry
angles
equal angles
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Source: 2010 Romania JBMO TST5 P2
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parmenides51
30630 posts
parmenides51
#1 h  May 25, 2020, 2:41 AM
Y by
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$
Z K Y
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badumtsss
61 posts
badumtsss
#2 h  May 25, 2020, 3:57 AM
Y by
$DF \parallel AC$, then $\angle DAC = \angle ABE \Longleftrightarrow \angle FDA = \angle FBE \Longleftrightarrow B,D,G,F$ is cyclic, where $G$ is the intersection of $AD,BE,CF$. We see $ B,D,G,F$ is cyclic $\Longleftrightarrow \angle AFC = \angle BDA$
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