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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.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

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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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F.E....can you solve it?
Jackson0423   14
N 16 minutes ago by maromex
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x \]for all real numbers \( x \) satisfying \( f(x) \neq 1 \).
14 replies
Jackson0423
3 hours ago
maromex
16 minutes ago
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Benelux fe
ErTeeEs06   11
N 17 minutes ago by Pitchu-25
Source: BxMO 2025 P1
Does there exist a function $f:\mathbb{R}\to \mathbb{R}$ such that $$f(x^2+f(y))=f(x)^2-y$$for all $x, y\in \mathbb{R}$?
11 replies
1 viewing
ErTeeEs06
Apr 26, 2025
Pitchu-25
17 minutes ago
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Consecutive averages of sequence always integer squares
math154   40
N 18 minutes ago by cursed_tangent1434
Source: USA December TST for IMO 2014, Problem 2
Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square).

Evan O'Dorney and Victor Wang
40 replies
+1 w
math154
Dec 24, 2013
cursed_tangent1434
18 minutes ago
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Problem3
samithayohan   115
N 23 minutes ago by bjump
Source: IMO 2015 problem 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Proposed by Ukraine
115 replies
samithayohan
Jul 10, 2015
bjump
23 minutes ago
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hard problem
Cobedangiu   0
26 minutes ago
$a,b,c>0$ and $a+b+c=7$. CM:
$\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+abc \ge ab+bc+ca-2$
0 replies
Cobedangiu
26 minutes ago
0 replies
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IMO 2008, Question 4
orl   119
N 38 minutes ago by lpieleanu
Source: IMO Shortlist 2008, A1
Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that
\[ \frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2} \]
for all positive real numbers $ w,x,y,z,$ satisfying $ wx = yz.$


Author: Hojoo Lee, South Korea
119 replies
1 viewing
orl
Jul 17, 2008
lpieleanu
38 minutes ago
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P29 [Geometry] - Turkish NMO 1st Round - 2004
matematikolimpiyati   2
N 39 minutes ago by ehuseyinyigit
Let $M$ be the intersection of the diagonals $AC$ and $BD$ of cyclic quadrilateral $ABCD$. If $|AB|=5$, $|CD|=3$, and $m(\widehat{AMB}) = 60^\circ$, what is the circumradius of the quadrilateral?

$ \textbf{(A)}\ 5\sqrt 3 \qquad\textbf{(B)}\ \dfrac {7\sqrt 3}{3} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{34} $
2 replies
matematikolimpiyati
Nov 12, 2013
ehuseyinyigit
39 minutes ago
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IMO Genre Predictions
ohiorizzler1434   45
N 41 minutes ago by ehuseyinyigit
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
45 replies
ohiorizzler1434
May 3, 2025
ehuseyinyigit
41 minutes ago
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2f(f(xy)-f(x+y))=xy+f(x)f(y)
dangerousliri   6
N 42 minutes ago by jasperE3
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that:
$$2f\left(f(xy)-f(x+y)\right)=xy+f(x)f(y)$$for all $x,y\in\mathbb{R}$.
6 replies
dangerousliri
May 12, 2019
jasperE3
42 minutes ago
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Numbers on cards (again!)
popcorn1   78
N an hour ago by maromex
Source: IMO 2021 P1
Let $n \geqslant 100$ be an integer. Ivan writes the numbers $n, n+1, \ldots, 2 n$ each on different cards. He then shuffles these $n+1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
78 replies
popcorn1
Jul 20, 2021
maromex
an hour ago
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Number Theory Chain!
JetFire008   61
N an hour ago by JetFire008
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
61 replies
JetFire008
Apr 7, 2025
JetFire008
an hour ago
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Algebra Pure one
Jackson0423   0
an hour ago

Let \( x_1, x_2, \dots, x_6 \) be six distinct real numbers satisfying the following condition:

For each \( i = 1, 2, \dots, 6 \),
\[ x_i^6 + (-1)^{i+1} x_i = -x_1 x_2 x_3 x_4 x_5 x_6. \]
Find the maximum value of \( x_1 x_2 + x_3 x_4 + x_5 x_6 \).
0 replies
Jackson0423
an hour ago
0 replies
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Number theory
MathsII-enjoy   0
an hour ago
Prove that when $x^p+y^p$ | $(p^2-1)^n$ with $x,y$ are positive integers and $p$ is prime ($p>3$), we get: $x=y$
0 replies
MathsII-enjoy
an hour ago
0 replies
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CooL geo
Pomegranat   1
N 2 hours ago by Pomegranat
Source: Idk

In triangle \( ABC \), \( D \) is the midpoint of \( BC \). \( E \) is an arbitrary point on \( AC \). Let \( S \) be the intersection of \( AD \) and \( BE \). The line \( CS \) intersects with the circumcircle of \( ACD \), for the second time at \( K \). \( P \) is the circumcenter of triangle \( ABE \). Prove that \( PK \perp CK \).
1 reply
1 viewing
Pomegranat
Today at 5:57 AM
Pomegranat
2 hours ago
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A problem about sum of all digits of a number
Tung-CHL   1
N Apr 25, 2025 by Tung-CHL
Source: Problems from the book
We say that a number $n$ is good if $n=m+s(m)$ for some positive integer $m$. Here $s(m)$ is the sum of all digits of $m$. Prove that:
$$10^k+n\; \text{is good for infinitely many} \; k \Longleftrightarrow n-1 \; \text{is good}$$
1 reply
Tung-CHL
Jul 16, 2022
Tung-CHL
Apr 25, 2025
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A problem about sum of all digits of a number
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Reveal topic tags
number theory
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Source: Problems from the book
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Tung-CHL
126 posts
Tung-CHL
#1 h  Jul 16, 2022, 8:48 AM
Y by
We say that a number $n$ is good if $n=m+s(m)$ for some positive integer $m$. Here $s(m)$ is the sum of all digits of $m$. Prove that:
$$10^k+n\; \text{is good for infinitely many} \; k \Longleftrightarrow n-1 \; \text{is good}$$
Z K Y
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Tung-CHL
126 posts
Tung-CHL
#2 h  Apr 25, 2025, 2:26 PM
Y by
This is actually not true!

There are infinitely many pairs of positive integers $(t,q)$ such that $9t=10^{q+1}+17$. Consider the number
$$m=99\dots900\dots02$$with $t$ digits $9$'s and $q$ digits $0$'s. We have
$$m+s(m)=m+9t+2=m+10^{q+1}+17+2=100\dots021=10^k+21$$where $k=t+q+1$. This means there are infinitely many $k$ such that $10^k+21$ is good. But $21-1=20$ is not a good number.
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