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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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b+c <=a/sin(A/2)
lgx57   1
N 14 minutes ago by sqing
Prove that: In $\triangle ABC$,$b+c \le \dfrac{a}{\sin \frac{A}{2}}$
1 reply
lgx57
an hour ago
sqing
14 minutes ago
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Inequalities
sqing   7
N 27 minutes ago by sqing
Let $ a,b,c>0 , a+b+c +abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$Let $ a,b,c>0 , ab+bc+ca+abc=4$. Prove that
$$ \frac {a}{a^2+2}+\frac {b}{b^2+2}+\frac {c}{c^2+2} \leq 1$$
7 replies
+1 w
sqing
Yesterday at 12:28 PM
sqing
27 minutes ago
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2024 Mock AIME 1 ** p15 (cheaters' trap) - 128 | n^{\sigma (n)} - \sigma(n^n)
parmenides51   5
N an hour ago by Amkan2022
Let $N$ be the number of positive integers $n$ such that $n$ divides $2024^{2024}$ and $128$ divides
$$n^{\sigma (n)} - \sigma(n^n)$$where $\sigma (n)$ denotes the number of positive integers that divide $n$, including $1$ and $n$. Find the remainder when $N$ is divided by $1000$.
5 replies
parmenides51
Jan 29, 2025
Amkan2022
an hour ago
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concyclic wanted, diameter related
parmenides51   3
N an hour ago by Giant_PT
Source: China Northern MO 2023 p1 CNMO
As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle.
IMAGE
3 replies
parmenides51
May 5, 2024
Giant_PT
an hour ago
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Concurrency
Omid Hatami   14
N an hour ago by Ilikeminecraft
Source: Iran TST 2008
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.
14 replies
Omid Hatami
May 20, 2008
Ilikeminecraft
an hour ago
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<ACB=90^o if AD = BD , <ACD = 3 <BAC, AM=//MD, CM//AB,
parmenides51   2
N an hour ago by AylyGayypow009
Source: 2021 JBMO TST Bosnia and Herzegovina P3
In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.
2 replies
parmenides51
Oct 7, 2022
AylyGayypow009
an hour ago
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Good Permutations in Modulo n
swynca   9
N an hour ago by optimusprime154
Source: BMO 2025 P1
An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that:
$(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$;
$(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$.
Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.
9 replies
swynca
Apr 27, 2025
optimusprime154
an hour ago
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Grid combo with tilings
a_507_bc   7
N 2 hours ago by john0512
Source: All-Russian MO 2023 Final stage 10.6
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?
7 replies
a_507_bc
Apr 23, 2023
john0512
2 hours ago
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sqrt(2)<=|1+z|+|1+z^2|<=4
SuiePaprude   3
N 2 hours ago by alpha31415
let z be a complex number with |z|=1 show that sqrt2 <=|1+z|+|1+z^2|<=4
3 replies
SuiePaprude
Jan 23, 2025
alpha31415
2 hours ago
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Simple but hard
Lukariman   5
N 2 hours ago by Giant_PT
Given triangle ABC. Outside the triangle, construct rectangles ACDE and BCFG with equal areas. Let M be the midpoint of DF. Prove that CM passes through the center of the circle circumscribing triangle ABC.
5 replies
Lukariman
Today at 2:47 AM
Giant_PT
2 hours ago
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inequalities
Ducksohappi   0
2 hours ago
let a,b,c be non-negative numbers such that ab+bc+ca>0. Prove:
$ \sum_{cyc} \frac{b+c}{2a^2+bc}\ge \frac{6}{a+b+c}$
P/s: I have analysed:$ S_a=\frac{b^2+c^2+3bc-ab-ac}{(2b^2+ac)(2c^2+2ab)}$, similarly to $S_b, S_c$, by SOS
0 replies
Ducksohappi
2 hours ago
0 replies
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bulgarian concurrency, parallelograms and midpoints related
parmenides51   7
N 2 hours ago by Ilikeminecraft
Source: Bulgaria NMO 2015 p5
In a triangle $\triangle ABC$ points $L, P$ and $Q$ lie on the segments $AB, AC$ and $BC$, respectively, and are such that $PCQL$ is a parallelogram. The circle with center the midpoint $M$ of the segment $AB$ and radius $CM$ and the circle of diameter $CL$ intersect for the second time at the point $T$. Prove that the lines $AQ, BP$ and $LT$ intersect in a point.
7 replies
parmenides51
May 28, 2019
Ilikeminecraft
2 hours ago
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Interesting inequalities
sqing   3
N 2 hours ago by sqing
Source: Own
Let $a,b,c \geq 0 $ and $ab+bc+ca- abc =3.$ Show that
$$a+k(b+c)\geq 2\sqrt{3 k}$$Where $ k\geq 1. $
Let $a,b,c \geq 0 $ and $2(ab+bc+ca)- abc =31.$ Show that
$$a+k(b+c)\geq \sqrt{62k}$$Where $ k\geq 1. $
3 replies
sqing
Today at 4:34 AM
sqing
2 hours ago
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If $a\cos A+b\sin A=m,$ and $a\sin A-b\cos A=n,$ then find the value of $a^2 +b^
Vulch   1
N 5 hours ago by Captainscrubz
If $a\cos A+b\sin A=m,$ and $a\sin A-b\cos A=n,$ then find the value of $a^2 +b^2.$
1 reply
Vulch
Today at 7:54 AM
Captainscrubz
5 hours ago
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Functional Equation
ab_xy123   4
N Apr 1, 2025 by millennium2k
Find all solutions to the functional equation $f(1-x) = f(x) + 1 - 2x$
4 replies
ab_xy123
Mar 16, 2020
millennium2k
Apr 1, 2025
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Functional Equation
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Reveal topic tags
algebra
functional equation
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ab_xy123
83 posts
ab_xy123
#1 h  Mar 16, 2020, 1:08 PM
Y by
Find all solutions to the functional equation $f(1-x) = f(x) + 1 - 2x$
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ab_xy123
83 posts
ab_xy123
#2 h  Mar 16, 2020, 2:38 PM • 1 Y
Y by Mango247
bumppp.....
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vvluo
1574 posts
vvluo
#3 h  Mar 16, 2020, 3:26 PM • 1 Y
Y by FaithnHope
This is a(rather difficult) problem from the Intermediate Algebra book.
The key to solving the problem is to see that $f(1-x)-(1-x) = f(x)-x$.
This is helpful because it allows us to substitute $g(x)=f(x)-x$ into the equation.
Doing this, we have $g(1-x)=g(x)$.
What this tells us is that $g(x)$ is symmetric across $x=\frac{1}{2}$.
We can now express $g(x)$ as a function symmetrical about the y-axis $(x=0)$ shifted to the right by $\frac{1}{2}$.
Thus $g(x)=h(x-\frac{1}{2})$ for some even(symmetric about the y-axis) function $h(x)$.
Therefore, $f(x)=h(x-\frac{1}{2}) +x$ for any even function $h(x)$.
800th post :)
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jasperE3
11346 posts
jasperE3
#4 h  Mar 31, 2025, 6:30 AM
Y by
ab_xy123 wrote:
Find all solutions to the functional equation $f(1-x) = f(x) + 1 - 2x$

General form is to choose any function $g:[0,\infty)\to\mathbb R$, then $f(x)=g\left|x-\frac12\right|+x$.
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millennium2k
1 post
millennium2k
#5 h  Apr 1, 2025, 10:35 AM
Y by
To solve the problem, replace x to 1-x, then solve the system of the equation!
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