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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Yesterday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

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Interested to learn more about our WOOT programs? Check out the course page here or join a Free Scheduled Info Session. Early bird pricing ends August 19th!:
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0 replies
jwelsh
Yesterday at 2:14 PM
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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Game with polynomials
old_csk_mo   1
N 2 minutes ago by sarjinius
Source: CAPS 2025 p3
Maryam and Artur play a game on a board, taking turns. At the beginning, the polynomial $XY-1$ is written on the board. Artur is the first to make a move. In each move, the player replaces the polynomial $P(X,Y)$ on the board with one of the following polynomials of their choice:
(a) $X\cdot P(X,Y),$
(b) $Y\cdot P(X,Y),$
(c) $P(X,Y)+a,$ where $a\le 2025$ is an arbitrary integer.
The game stops after each player has made 2025 moves. Let $Q(X,Y)$ be the polynomial on the board after the game ends. Maryam wins if the equation $Q(x,y)=0$ has a finite and odd number of positive integer solutions $(x,y).$ Show that Maryam can always win the game, no matter how Artur plays.
1 reply
old_csk_mo
Jul 27, 2025
sarjinius
2 minutes ago
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FEs are still alive!
anantmudgal09   3
N 15 minutes ago by kan_ari
Source: India-Iran-Singapore-Taiwan Friendly Contest 2025 Problem 4
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xf(y) + y^2) = (x + y)f(x) + (x + f(y))f(y - x)$$for all $x, y \in \mathbb{R}$.
3 replies
anantmudgal09
Today at 7:17 AM
kan_ari
15 minutes ago
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Geometry Problem
Hopeooooo   13
N 21 minutes ago by kotmhn
Source: SRMC 2022 P1
Convex quadrilateral $ABCD$ is inscribed in circle $w.$Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$
(Kungozhin M.)
13 replies
Hopeooooo
May 23, 2022
kotmhn
21 minutes ago
J
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IMO ShortList 1999, geometry problem 8
orl   23
N 33 minutes ago by YaoAOPS
Source: IMO ShortList 1999, geometry problem 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
23 replies
orl
Nov 13, 2004
YaoAOPS
33 minutes ago
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Maximize non-intersecting/perpendicular diagonals!
cjquines0   38
N 37 minutes ago by lpieleanu
Source: 2016 IMO Shortlist C5
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
38 replies
cjquines0
Jul 19, 2017
lpieleanu
37 minutes ago
J
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Albanian Junior Math Contest question
Deomad123   4
N 41 minutes ago by P0tat0b0y
Show that for all $n \in \mathbb{N}$ the inequality holds: $\frac{1}{2}<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}<1$.
4 replies
Deomad123
Jul 30, 2025
P0tat0b0y
41 minutes ago
J
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Wild-looking multi-set algebra
anantmudgal09   1
N an hour ago by L567
Source: India-Iran-Singapore-Taiwan Friendly Contest 2025 Problem 3
For a multiset $A$, define $$f(A, i, m) = \sum_{a \in A, \, 3 \mid a-i} a^m$$and let $g(A, m)$ be the set $\{f(A, 0, m), f(A, 1, m), f(A, 2, m)\}$.

Suppose for some multi-set $S$ we have that $$\left|g(S, 0)\right|=\left|g(S, 1)\right|=1, \left|g(S, 2)\right|=3.$$
Prove that there exists some integer $k \ne 0$ divisible by $6$ such that if we define multi-set, $T := \{x_1+x_2+\dots+x_k \, | \, (x_1, x_2, \dots, x_k) \in S^k\}$ then $$f(T, 0, 2k) \leqslant \frac{f(T, 1, 2k)+f(T, 2, 2k)}{2}.$$
1 reply
1 viewing
anantmudgal09
Today at 7:16 AM
L567
an hour ago
J
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Equal segments with humpty and dumpty points.
Kratsneb   1
N an hour ago by aqwxderf
Let $X$, $Y$ be such points on sides $AB$, $AC$ of a triangle $ABC$ that $BXYC$ is cyclic. $BY \cap CX = D$, $N$ is the midpoint of $AD$. In triangles $BDX$ and $CDY$ let $P$, $Q$ be the $D$-humpty points and let $S$, $T$ be the $D$-dumpty points. Prove that $AP = AQ$ and $NS = NT$.
IMAGE
1 reply
Kratsneb
4 hours ago
aqwxderf
an hour ago
J
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Multivariate polynomial must vanish on all permutations
DottedCaculator   2
N an hour ago by CANBANKAN
Source: 2025 ELMO Shortlist A4
Fix positive integers $n$ and $k$ with $n \geq k$. Determine the smallest positive integer $d$ satisfying the following condition:

For any (not necessarily distinct) real numbers $a_1$, $\dots$, $a_n$, there exists a real-coefficient polynomial $f$ in $k$ variables satisfying the following properties:
[list]
[*] $f$ has degree at most $d$;
[*] $f$ is not identically $0$;
[*] for any permutation $b_1, \ldots, b_n$ of $a_1, \ldots, a_n$, the equation $f(b_1, \ldots, b_k) = 0$ holds.
[/list]

Benny Wang
2 replies
DottedCaculator
Jun 30, 2025
CANBANKAN
an hour ago
J
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Combinatorics
slimshady360   0
an hour ago
Source: IMO 2005-Problem 6.
Does anybody have any idea how to solve this with graph theory?
0 replies
slimshady360
an hour ago
0 replies
J
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USAMO 2003 Problem 1
MithsApprentice   75
N 2 hours ago by SomeonecoolLovesMaths
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
75 replies
MithsApprentice
Sep 27, 2005
SomeonecoolLovesMaths
2 hours ago
J
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Some of my less-seen proposals
navid   7
N 2 hours ago by shaboon
Dear friends,

Since 2003, I have had several nice days in AOPS-- i.e., Mathlinks; as some of you may remember. I decided to share you some of my less-seen proposals. Some of them may be considered as some early ethudes; several of them already appeared on some competitions or journals. I hope you like them and this be a good starting point for working on them. Please take a look at the following link.

https://drive.google.com/file/d/1bntcjZAHZ-WN1lfGbNbz0uyFvhBTMEhz/view?usp=sharing

Best regards,
Navid.
7 replies
navid
Jul 30, 2025
shaboon
2 hours ago
J
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Ugly functional equation
Taha1381   11
N 2 hours ago by shaboon
Source: Iranian third round 2019 Finals algebra exam problem 3
Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that:

$af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$

For all positive real $x$ and large enough $y$.

Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that:

$f(xy)+f(\frac{x}{y})=2f(x)+h(y)$

For all positive real $x$ and large enough $y$.
11 replies
Taha1381
Aug 18, 2019
shaboon
2 hours ago
J
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Bicentric Quadrilateral Concurrence
anantmudgal09   2
N 2 hours ago by MathLuis
Source: India-Iran-Singapore-Taiwan Friendly Contest 2025 Problem 2
Let $ABCD$ be a quadrilateral with both an incircle and a circumcircle. Let $I$ and $O$ be the incenter and circumcenter of $ABCD$, respectively. Let $E$ be the intersection of lines $AB$ and $CD$, and let $F$ be the intersection of lines $BC$ and $DA$. Let $X$ and $Y$ be the intersections of the line $FI$ with lines $AB$ and $CD$, respectively. Prove that the circumcircle of $\triangle EIF$, the circumcircle of $\triangle EXY$, and the line $FO$ are concurrent.
2 replies
anantmudgal09
Today at 7:15 AM
MathLuis
2 hours ago
J
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J
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Combinatorics Problem
P.J   8
N May 14, 2025 by MITDragon
Source: Mexican Mathematical Olympiad Problems Book
Calculate the sum of 1 x 1000 + 2 x 999 + ... + 999 x 2 + 1000 x 1
8 replies
P.J
Dec 28, 2024
MITDragon
May 14, 2025
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Combinatorics Problem
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Reveal topic tags
combinatorics
L
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Source: Mexican Mathematical Olympiad Problems Book
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P.J
6 posts
P.J
#1 h  Dec 28, 2024, 10:16 AM
Y by
Calculate the sum of 1 x 1000 + 2 x 999 + ... + 999 x 2 + 1000 x 1
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OSAGHO
2 posts
OSAGHO
#2 h  Dec 28, 2024, 10:59 AM
Y by
Kindly correct me if i am wrong
the answer i have come up wih - 167167500
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Sir_Numbercrunch
176 posts
Sir_Numbercrunch
#3 h  Dec 28, 2024, 12:09 PM
Y by
you are right
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Just1
74 posts
Just1
#4 h  Dec 28, 2024, 4:58 PM • 1 Y
Y by P.J
Hello but I got answer 167167000.I showed each term of sum like x*(1001-x) then I got the whole sum like this: 1001*(1+2+..+1000)-(1^2+2^2+...+1000^2) so my answer is 1001*334*500 which is equal to 167167000
If I did smth wrong then please,point it polite:)
This post has been edited 1 time. Last edited by Just1, Dec 28, 2024, 5:02 PM
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Sir_Numbercrunch
176 posts
Sir_Numbercrunch
#5 h  Dec 28, 2024, 5:01 PM • 1 Y
Y by P.J
Sir_Numbercrunch wrote:
you are (almost) right
Thanks @below I didn't realize our answers differed by 500.
Z K Y
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Just1
74 posts
Just1
#6 h  Dec 28, 2024, 5:07 PM
Y by
but what is my fault?
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P.J
6 posts
P.J
#7 h  Dec 29, 2024, 4:58 PM
Y by
#4 & #5 Impressive
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Ianis
447 posts
Ianis
#8 h  Dec 29, 2024, 5:32 PM
Y by
In general the sum is\begin{align*}\sum \limits _{k=1}^nk(n+1-k) & =\sum \limits _{k=1}^nk(n+1)-\sum \limits _{k=1}^nk^2 \\ & =(n+1)\frac{n(n+1)}{2}-\frac{n(n+1)(2n+1)}{6} \\ & =n(n+1)\left (\frac{n+1}{2}-\frac{2n+1}{6}\right ) \\ & =\frac{n(n+1)(n+2)}{6} \\ & =\binom{n+2}{3}. \end{align*}When $n=1000$ we get $167167000$.
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MITDragon
18 posts
MITDragon
#9 h  May 14, 2025, 12:58 PM
Y by
This can be rewritten as:
\[ \sum_{k=1}^{1000} k \cdot (1001 - k) \]
Expanding the expression:
\[ \sum_{k=1}^{1000} \left(1001k - k^2\right) = 1001 \sum_{k=1}^{1000} k - \sum_{k=1}^{1000} k^2 \]
Use the formulas for the sum of the first \(n\) natural numbers and the sum of their squares:
\[ \sum_{k=1}^{n} k = \frac{n(n+1)}{2}, \quad \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]
For \(n = 1000\):
\[ \sum_{k=1}^{1000} k = \frac{1000 \cdot 1001}{2} = 500500 \]\[ \sum_{k=1}^{1000} k^2 = \frac{1000 \cdot 1001 \cdot 2001}{6} = 333833500 \]
Therefore:
\[ 1001 \cdot 500500 - 333833500 = 501000500 - 333833500 = \boxed{167167000} \]
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