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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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Inspired by old results
sqing   1
N 7 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a+b+c =2.$ Prove that
$$ a b+b c +c a+ a^2b^2+b^2c^2+c^2a^2+\frac{1}{4} a b c \leq2$$$$a b+b c +c a+ a^3b^3+b^3c^3+c^3a^3+\frac{49}{36} a b c \leq2$$$$ a b+b c +c a+ a^4b^4+b^4c^4+c^4a^4+\frac{601}{324} \leq2$$
1 reply
2 viewing
sqing
43 minutes ago
sqing
7 minutes ago
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Find the value
sqing   7
N 7 minutes ago by Phat_23000245
Source: 2024 China Fujian High School Mathematics Competition
Let $f(x)=a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0,$ $a_i\in\{-1,1\} ,i=0,1,2,\cdots,6 $ and $f(2)=-53 .$ Find the value of $f(1).$
7 replies
sqing
Jun 22, 2024
Phat_23000245
7 minutes ago
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IMO ShortList 2002, algebra problem 2
orl   28
N 9 minutes ago by ezpotd
Source: IMO ShortList 2002, algebra problem 2
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that \[\left\lvert a_i-a_j \right\rvert\geq \frac{1}{i+j} \quad \text{for all }i,\ j \text{ with } i \neq j. \] Prove that $c\geq1$.
28 replies
orl
Sep 28, 2004
ezpotd
9 minutes ago
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Erasing the difference of two numbers
BR1F1SZ   3
N 21 minutes ago by wizardvexik
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.)

For which values of $n$ is it possible to end with only one number remaining on the board?

(Michael Reitmeir)
3 replies
+1 w
BR1F1SZ
May 5, 2025
wizardvexik
21 minutes ago
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sum of gcd over sets is more then sum of gcd over union
Miquel-point   2
N an hour ago by N3bula
Source: KoMaL A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]Proposed by Dávid Matolcsi, Berkeley
2 replies
Miquel-point
Jun 11, 2024
N3bula
an hour ago
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Interesting inequalities
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$$a(bc+bd+cd) \leq \frac{256}{81}$$$$ ab(a+2c+2d ) \leq \frac{256}{27}$$$$ ab(a+3c+3d ) \leq \frac{32}{3}$$$$ ab(c+d ) \leq \frac{64}{27}$$
2 replies
sqing
Yesterday at 1:25 PM
sqing
an hour ago
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A sharp one with 3 var
mihaig   5
N 2 hours ago by mihaig
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$ab+bc+ca+abc\geq4.$$
5 replies
mihaig
May 13, 2025
mihaig
2 hours ago
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3 var inequality
JARP091   7
N 2 hours ago by MathsII-enjoy
Source: Own
Let \( x, y, z \in \mathbb{R}^+ \). Prove that
\[ \sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2} \]without using the Rearrangement Inequality or Chebyshev's Inequality.
7 replies
JARP091
Yesterday at 8:54 AM
MathsII-enjoy
2 hours ago
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System of Equations
Math-wiz   18
N 2 hours ago by justaguy_69
Source: RMO 2019 Paper 2 P3
Find all triples of non-negative real numbers $(a,b,c)$ which satisfy the following set of equations
$$a^2+ab=c$$$$b^2+bc=a$$$$c^2+ca=b$$
18 replies
1 viewing
Math-wiz
Nov 10, 2019
justaguy_69
2 hours ago
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Geometry
MathsII-enjoy   5
N 2 hours ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$ with $M$ being the midpoint of $BC$. The tangents at $B, C$ of $(O)$ intersect at $D$. Let $N$ be the projection of $O$ onto $AD$. On the perpendicular bisector of $BC$, take a point $K$ that is not on $(O)$ and different from M. Circle $(KBC)$ intersects $AK$ at $F$. Lines $NF$ and $AM$ intersect at $E$. Prove that $AEF$ is an isosceles triangle.
5 replies
MathsII-enjoy
May 15, 2025
MathsII-enjoy
2 hours ago
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Self-evident inequality trick
Lukaluce   18
N 2 hours ago by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let $x, y$, and $z$ be positive real numbers, such that $x^2 + y^2 + z^2 = 3$. Prove the inequality
\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]When does the equality hold?
18 replies
Lukaluce
May 18, 2025
sqing
2 hours ago
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Cauchy FE
InftyByond   1
N 3 hours ago by maromex
For the Cauchy FE, is x^nf(x)=f(x^(n+1)) enouh to go from rationals to reals? thats what wikipedia says but idk really
1 reply
InftyByond
Mar 1, 2025
maromex
3 hours ago
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Train yourself on folklore NT FE ideas
Assassino9931   3
N 3 hours ago by MathLuis
Source: Bulgaria Spring Mathematical Competition 2025 9.4
Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.
3 replies
Assassino9931
Mar 30, 2025
MathLuis
3 hours ago
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IMO ShortList 1998, number theory problem 2
orl   16
N 3 hours ago by littlefox_amc
Source: IMO ShortList 1998, number theory problem 2
Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)
16 replies
orl
Oct 22, 2004
littlefox_amc
3 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
J
A problem about sum of all digits of a number
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Reveal topic tags
number theory
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G H BBookmark kLocked kLocked NReply
Source: Problems from the book
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Tung-CHL
129 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
129 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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