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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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[*]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
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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2025 Guangdong High School Mathematics Competition Q14
sqing   0
25 minutes ago
Source: China
Let $ x_1, x_2, x_3, x_4, x_5\geq 0 $ and $ x^2_1+x^2_2+x^2_3+ x^2_4+ x^2_5=4. $ Find the maximum value of
$\sum_{i=1}^5 \frac{1}{x_i+1} \sum_{i=1}^5 x_i .$
0 replies
1 viewing
sqing
25 minutes ago
0 replies
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Nice "if and only if" function problem
ICE_CNME_4   5
N 28 minutes ago by wh0nix
Let $f : [0, \infty) \to [0, \infty)$, $f(x) = \dfrac{ax + b}{cx + d}$, with $a, d \in (0, \infty)$, $b, c \in [0, \infty)$. Prove that there exists $n \in \mathbb{N}^*$ such that for every $x \geq 0$
\[ f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0. \](For $n \in \mathbb{N}^*$ and $x \geq 0$, the notation $f_n(x)$ represents $\underbrace{(f \circ f \circ \dots \circ f)}_{n \text{ times}}(x)$. )

Please do it at 9th grade level. Thank you!
5 replies
ICE_CNME_4
Yesterday at 7:23 PM
wh0nix
28 minutes ago
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2025 Xinjiang High School Mathematics Competition Q11
sqing   0
33 minutes ago
Source: China
Let $ a,b,c >0 . $ Prove that
$$ \left(1+\frac {a} { b}\right)\left(1+\frac {a}{ b}+\frac {b}{ c}\right) \left(1+\frac {a}{b}+\frac {b}{ c}+\frac {c}{ a}\right) \geq 16 $$
0 replies
sqing
33 minutes ago
0 replies
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Cauchy-Schwarz 2
prtoi   7
N 35 minutes ago by mpcnotnpc
Source: Handout by Samin Riasat
if $a^2+b^2+c^2+d^2=4$, prove that:
$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\ge4$
7 replies
prtoi
Mar 26, 2025
mpcnotnpc
35 minutes ago
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Japanese Olympiad
parkjungmin   7
N an hour ago by Gauler
It's about the Japanese Olympiad

I can't solve it no matter how much I think about it.

If there are people who are good at math

Please help me.
7 replies
parkjungmin
May 10, 2025
Gauler
an hour ago
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Center lies on circumcircle of other
Philomath_314   40
N an hour ago by lakshya2009
Source: INMO P1
In triangle $ABC$ with $CA=CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$.

Proposed by Prithwijit De
40 replies
Philomath_314
Jan 21, 2024
lakshya2009
an hour ago
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Rearrangement
Jackson0423   0
an hour ago

Let \( a_1, a_2, \ldots, a_{2025} \) be a rearrangement of the integers from 1 to 2025.
For each \( i \), define \( c_i \) to be the number of integers \( j \) such that \( 1 \leq j < i \leq 2025 \) and \( a_j > a_i \).
Suppose non-negative integers \( x_1, x_2, \ldots, x_{2025} \) are given such that \( c_i = x_i \) and \( x_i < i \) for all \( i \).
Prove that there exists a unique permutation \( (a_1, a_2, \ldots, a_{2025}) \) satisfying this condition.
0 replies
Jackson0423
an hour ago
0 replies
J
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Nice geometry
gggzul   2
N an hour ago by gggzul
Let $ABC$ be a acute triangle with $\angle BAC=60^{\circ}$. $H, O$ are the orthocenter and circumcenter. Let $D$ be a point on the same side of $OH$ as $A$, such that $HDO$ is equilateral. Let $P$ be a point on the same side of $BD$ as $A$, such that $BDP$ is equilateral. Let $Q$ be a point on the same side of $CD$ as $A$, such that $CDP$ is equilateral. Let $M$ be the midpoint of $AD$. Prove that $P, M, Q$ are collinear.
2 replies
gggzul
3 hours ago
gggzul
an hour ago
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Inspired by old results
sqing   2
N an hour ago by sqing
Source: Own
Let $ a, b> 0,a + 2b= 1. $ Prove that
$$ \sqrt{a + b^2} +2 \sqrt{b+ a^2} + |a - b| \geq 2$$Let $ a, b> 0,a + 2b= \frac{3}{4}. $ Prove that
$$ \sqrt{a + (b - \frac{1}{4})^2} +2 \sqrt{b + (a- \frac{1}{4})^2} + \sqrt{ (a - b)^2+ \frac{1}{4}} \geq 2$$
2 replies
sqing
May 20, 2025
sqing
an hour ago
J
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non-symmetric inequality
RainbowNeos   1
N an hour ago by RainbowNeos
Source: competition in Xinjiang, China
Given $a,b,c>0$, show that
\[\left(1+\frac{a}{b}\right)\left(1+\frac{a}{b}+\frac{b}{c}\right)\left(1+\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)\geq 16\]
1 reply
RainbowNeos
an hour ago
RainbowNeos
an hour ago
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Inspired by RMO 2006
sqing   0
2 hours ago
Source: Own
Let $ a,b >0 . $ Prove that
$$ \frac {a^{2}+1}{b+k}+\frac { b^{2}+1}{ka+1}+\frac {2}{a+kb} \geq \frac {6}{k+1} $$Where $k\geq 0.03 $
$$ \frac {a^{2}+1}{b+1}+\frac { b^{2}+1}{a+1}+\frac {2}{a+b} \geq 3 $$
0 replies
sqing
2 hours ago
0 replies
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Different Paths Probability
Qebehsenuef   3
N 3 hours ago by Etkan
Source: OBM
A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?
3 replies
Qebehsenuef
Apr 28, 2025
Etkan
3 hours ago
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Find the expected END time for the given process
superpi   2
N 4 hours ago by Hello_Kitty
This problem suddenly popped up in my head. But I don't know how to deal with it.

There are N bulbs. All the bulbs' available time follows same exponential distribution with parameter lambda(Or any arbitrary distribution with mean $\mu$). We do following operations
1. First, turn on the all $N$ bulbs
2. For each $k >= 2$ bulbs goes out, append ONE NEW BULB and turn on (This step starts and finishes immediately when kth bulb goes out)
3. Repeat 2 until all the bulbs goes out

What is the expected terminate time for the above process for given $N, k, \lambda$?

Or, is there any more conditions to complete the problem?

2 replies
superpi
Yesterday at 4:33 PM
Hello_Kitty
4 hours ago
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36x⁴ + 12x² - 36x + 13 > 0
fxandi   4
N Today at 10:22 AM by wh0nix
Prove that for any real $x \geq 0$ holds inequality $36x^4 + 12x^2 - 36x + 13 > 0.$
4 replies
fxandi
May 5, 2025
wh0nix
Today at 10:22 AM
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Matrix Row and column relation.
Schro   6
N May 1, 2025 by Schro
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
6 replies
Schro
Apr 28, 2025
Schro
May 1, 2025
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Matrix Row and column relation.
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Reveal topic tags
linear algebra
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Schro
4 posts
Schro
#1 h  Apr 28, 2025, 2:54 PM
Y by
If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...
Z K Y
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Samujjal101
2801 posts
Samujjal101
#2 h  Apr 28, 2025, 3:06 PM
Y by
What do you mean by dependent ?
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rchokler
2975 posts
rchokler
#3 h  Apr 28, 2025, 8:40 PM
Y by
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.
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Schro
4 posts
Schro
#4 h  Apr 29, 2025, 4:31 PM
Y by
Samujjal101 wrote:
What do you mean by dependent ?


I mean if ith row is a combination of other rows.....then ith column will also be a combination of other columns
Z K Y
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Schro
4 posts
Schro
#5 h  Apr 29, 2025, 4:32 PM
Y by
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa
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Etkan
1572 posts
Etkan
#6 h  Apr 29, 2025, 5:01 PM
Y by
Schro wrote:
rchokler wrote:
For square matrices, if the rows are linearly dependent, then so are the columns and vice-versa. This is because for every matrix of any size, the row rank and column rank are equal.

I m not talking in general sense....
I m being specific.....if "i th" row is a combination of other rows.....then "i th " column will be a combination of other columns and vice versa

It's not true. For example, in the matrix$$\begin{pmatrix}0&0&0\\1&0&0\\-1&0&0\end{pmatrix}$$the first row is the sum of the second and third rows, but any linear combination of the second and third columns is $0$, so it cannot be equal to the first column.
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Schro
4 posts
Schro
#7 h  May 1, 2025, 6:20 AM
Y by
Thnks, i got it
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