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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
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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functional equation with exponentials
produit   4
N 9 minutes ago by jasperE3
Find all solutions of the real valued functional equation:
f(\sqrt{x^2+y^2})=f(x)f(y).
Here we do not assume f is continuous
4 replies
produit
2 hours ago
jasperE3
9 minutes ago
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functional inequality with equality
miiirz30   1
N 11 minutes ago by MR.1
Source: 2025 Euler Olympiad, Round 2
Find all functions \( f : \mathbb{R} \to \mathbb{R} \) such that the following two conditions hold:

1. For all real numbers $a$ and $b$ satisfying $a^2 + b^2 = 1$, We have $f(x) + f(y) \geq f(ax + by)$ for all real numbers $x, y$.

2. For all real numbers $x$ and $y$, there exist real numbers $a$ and $b$, such that $a^2 + b^2 = 1$ and $f(x) + f(y) = f(ax + by)$.

Proposed by Zaza Melikidze, Georgia
1 reply
miiirz30
5 hours ago
MR.1
11 minutes ago
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Moving stones on an infinite row
miiirz30   1
N 14 minutes ago by genius_007
Source: 2025 Euler Olympiad, Round 2
We are given an infinite row of cells extending infinitely in both directions. Some cells contain one or more stones. The total number of stones is finite. At each move, the player performs one of the following three operations:

1. Take three stones from some cell, and add one stone to the cells located one cell to the left and one cell to the right, each skipping one cell in between.

2. Take two stones from some cell, and add one stone to the cell one cell to the left, skipping one cell and one stone to the adjacent cell to the right.

3. Take one stone from each of two adjacent cells, and add one stone to the cell to the right of these two cells.

The process ends when no moves are possible. Prove that the process always terminates and the final distribution of stones does not depend on the choices of moves made by the player.

IMAGE

Proposed by Luka Tsulaia, Georgia
1 reply
miiirz30
4 hours ago
genius_007
14 minutes ago
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Interesting inequalities
sqing   8
N 24 minutes ago by sqing
Source: Own
Let $ a,b,c,d\geq 0 , a+b+c+d \leq 4.$ Prove that
$$a(bc+bd+cd) \leq \frac{256}{81}$$$$ ab(a+2c+2d ) \leq \frac{256}{27}$$$$ ab(a+3c+3d ) \leq \frac{32}{3}$$$$ ab(c+d ) \leq \frac{64}{27}$$
8 replies
sqing
Yesterday at 1:25 PM
sqing
24 minutes ago
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2000 KJMO P1 easy euclidean lemma
RL_parkgong_0106   4
N 29 minutes ago by JH_K2IMO
Source: KJMO 2000
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
4 replies
RL_parkgong_0106
Jun 29, 2024
JH_K2IMO
29 minutes ago
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Sum floor 2^k/3 from k=0 to 100
v_Enhance   8
N 29 minutes ago by MathIQ.
Source: All-Russian MO 2000
Evaluate the sum \[ \left\lfloor \frac{2^0}{3} \right\rfloor + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor. \]
8 replies
v_Enhance
Dec 30, 2012
MathIQ.
29 minutes ago
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Points on a lattice path lies on a line
navi_09220114   3
N 30 minutes ago by Ianis
Source: TASIMO 2025 Day 1 Problem 3
Let $S$ be a nonempty subset of the points in the Cartesian plane such that for each $x\in S$ exactly one of $x+(0,1)$ or $x+(1,0)$ also belongs to $S$. Prove that for each positive integer $k$ there is a line in the plane (possibly different lines for different $k$) which contains at least $k$ points of $S$.
3 replies
navi_09220114
May 19, 2025
Ianis
30 minutes ago
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Graph Theory
ABCD1728   1
N 36 minutes ago by ABCD1728
Can anyone provide the PDF version of "Graphs: an introduction" by Radio Bumbacea (XYZ press), thanks!
1 reply
ABCD1728
Today at 5:32 AM
ABCD1728
36 minutes ago
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Inspired by Crux 4975
sqing   4
N an hour ago by sqing
Source: Own
Let $ a,b,c\geq 0 $ and $ a(b +c +bc)=1.$ Prove that
$$a^2+b^2+c^2+ 3abc(a+b+c)\geq 2 $$
4 replies
sqing
Oct 18, 2024
sqing
an hour ago
J
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CMI Entrance 19#6
bubu_2001   7
N an hour ago by Apple_maths60
$(a)$ Compute -
\begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg] \end{align*}$(b)$ For $x > 0 $ define $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $

$1.$ Determine the open interval(s) (if any) where $F ( x )$ is decreasing and all the open interval(s) (if any) where $F ( x )$ is increasing.

$2.$ Determine all the local minima of $F ( x )$ (if any) and all the local maxima of $F ( x )$ (if any) $.$
7 replies
bubu_2001
Nov 1, 2019
Apple_maths60
an hour ago
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How can I know the sequences's convergence value?
Madunglecha   5
N 4 hours ago by teomihai
What is the convergence value of the sequence??
(n^2)*ln(n+1/n)-n
5 replies
Madunglecha
Yesterday at 6:56 AM
teomihai
4 hours ago
J
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Confusion in derivative
Vulch   1
N 4 hours ago by Vulch
Solve the following problem:
1 reply
Vulch
4 hours ago
Vulch
4 hours ago
J
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Prove the statement
Butterfly   12
N 5 hours ago by oty
Given an infinite sequence $\{x_n\} \subseteq [0,1]$, there exists some constant $C$, for any $r>0$, among the sequence $x_n$ and $x_m$ could be chosen to satisfy $|n-m|\ge r $ and $|x_n-x_m|<\frac{C}{|n-m|}$.
12 replies
Butterfly
May 7, 2025
oty
5 hours ago
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Classifying Math with Symbols Based on Behavior
Midevilgmer   2
N Today at 4:06 AM by Midevilgmer
I have been working on a new math idea that prioritizes the behavior of numbers, equations, and expressions rather than their exact values. Numbers are described using symbols like P (Positive), N (Negative), Z (Zero), I (Imaginary), and D (Decimal). The goal is to create a system that uses symbols that allows you to perform operations like P*D and P+N and determine the behavioral outcomes based on the properties involved. For example, instead of identifying a number as 3, you would describe it as a positive, odd, whole, prime number, allowing you work with those traits individually or together. I would like to mention that I already have created an Addition, Subtraction, Multiplication, Division, Square Root, Exponent, and Factorial table that shows how these different behaviors work in basic operations. Finally, I want to mention that my current background includes a knowledge of geometry, algebra, and a very little amount of calculus. Any thoughts or ideas would be appreciated.
2 replies
Midevilgmer
Today at 12:55 AM
Midevilgmer
Today at 4:06 AM
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How to solve this problem
xiangovo   1
N Apr 21, 2025 by loup blanc
Source: website
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
1 reply
xiangovo
Mar 19, 2025
loup blanc
Apr 21, 2025
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How to solve this problem
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Reveal topic tags
algebraic geometry
superior algebra
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Source: website
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xiangovo
1 post
xiangovo
#1 h  Mar 19, 2025, 11:54 AM
Y by
How many nonzero points are there on x^3y + y^3z + z^3x = 0 over the finite field \mathbb{F}_{5^{18}} up to scaling?
Z K Y
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loup blanc
3599 posts
loup blanc
#2 h  Apr 21, 2025, 11:09 AM
Y by
With a computer, I find
for $r\leq 5$, the number of such points (up to scaling) over $\mathbb{F}_{5^r}$ is $5^r+1$.
Then I conjectured that the required result is $5^{18}+1$, but this is probably false. -see below-
EDIT 1. The considered projective curve is the Klein quartic over $\mathbb{F}_q$.
Indeed, the number of its points is $q+1$ except when $q=1 \mod 7$.
Unfortunately $5^r=1 \mod 7$ when $r=6,12,18,\cdots$.
EDIT 2. Over $\mathbb{F}_{5^6}$, the number of points is $16376$ and not $5^6+1=15626$.

I think that @ xiangovo is a joker.
This post has been edited 3 times. Last edited by loup blanc, Apr 21, 2025, 5:55 PM
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