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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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USAMO 2001 Problem 2
MithsApprentice   53
N 32 minutes ago by lksb
Let $ABC$ be a triangle and let $\omega$ be its incircle. Denote by $D_1$ and $E_1$ the points where $\omega$ is tangent to sides $BC$ and $AC$, respectively. Denote by $D_2$ and $E_2$ the points on sides $BC$ and $AC$, respectively, such that $CD_2=BD_1$ and $CE_2=AE_1$, and denote by $P$ the point of intersection of segments $AD_2$ and $BE_2$. Circle $\omega$ intersects segment $AD_2$ at two points, the closer of which to the vertex $A$ is denoted by $Q$. Prove that $AQ=D_2P$.
53 replies
MithsApprentice
Sep 30, 2005
lksb
32 minutes ago
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A=b
k2c901_1   89
N 40 minutes ago by reni_wee
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.

Proposed by Mohsen Jamali, Iran
89 replies
k2c901_1
Mar 29, 2006
reni_wee
40 minutes ago
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Strange angle condition and concyclic points
lminsl   129
N 43 minutes ago by Aiden-1089
Source: IMO 2019 Problem 2
In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$.

Prove that points $P,Q,P_1$, and $Q_1$ are concyclic.

Proposed by Anton Trygub, Ukraine
129 replies
1 viewing
lminsl
Jul 16, 2019
Aiden-1089
43 minutes ago
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Simple inequality
sqing   12
N an hour ago by Rayvhs
Source: MEMO 2018 T1
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1.$ Prove that$$\frac{a^2-b^2}{a+bc}+\frac{b^2-c^2}{b+ca}+\frac{c^2-a^2}{c+ab}\leq a+b+c-3.$$
12 replies
sqing
Sep 2, 2018
Rayvhs
an hour ago
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Random concyclicity in a square config
Maths_VC   2
N an hour ago by Maths_VC
Source: Serbia JBMO TST 2025, Problem 1
Let $M$ be a random point on the smaller arc $AB$ of the circumcircle of square $ABCD$, and let $N$ be the intersection point of segments $AC$ and $DM$. The feet of the tangents from point $D$ to the circumcircle of the triangle $OMN$ are $P$ and $Q$ , where $O$ is the center of the square. Prove that points $A$, $C$, $P$ and $Q$ lie on a single circle.
2 replies
Maths_VC
2 hours ago
Maths_VC
an hour ago
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Serbian selection contest for the IMO 2025 - P3
OgnjenTesic   3
N an hour ago by atdaotlohbh
Source: Serbian selection contest for the IMO 2025
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that:
- $f$ is strictly increasing,
- there exists $M \in \mathbb{N}$ such that $f(x+1) - f(x) < M$ for all $x \in \mathbb{N}$,
- for every $x \in \mathbb{Z}$, there exists $y \in \mathbb{Z}$ such that
\[ f(y) = \frac{f(x) + f(x + 2024)}{2}. \]Proposed by Pavle Martinović
3 replies
OgnjenTesic
May 22, 2025
atdaotlohbh
an hour ago
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Easy P4 combi game with nt flavour
Maths_VC   0
an hour ago
Source: Serbia JBMO TST 2025, Problem 4
Two players, Alice and Bob, play the following game, taking turns. In the beginning, the number $1$ is written on the board. A move consists of adding either $1$, $2$ or $3$ to the number written on the board, but only if the chosen number is coprime with the current number (for example, if the current number is $10$, then in a move a player can't choose the number $2$, but he can choose either $1$ or $3$). The player who first writes a perfect square on the board loses. Prove that one of the players has a winning strategy and determine who wins in the game.
0 replies
Maths_VC
an hour ago
0 replies
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USAMO 2003 Problem 1
MithsApprentice   70
N an hour ago by endless_abyss
Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.
70 replies
MithsApprentice
Sep 27, 2005
endless_abyss
an hour ago
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Inequalities
sqing   26
N an hour ago by ytChen
Let $ a,b>0 $ . Prove that
$$ \frac{a}{a^2+a +2b+1}+ \frac{b}{b^2+2a +b+1} \leq \frac{2}{5} $$$$ \frac{a}{a^2+2a +b+1}+ \frac{b}{b^2+a +2b+1} \leq \frac{2}{5} $$
26 replies
sqing
May 13, 2025
ytChen
an hour ago
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Basic ideas in junior diophantine equations
Maths_VC   0
an hour ago
Source: Serbia JBMO TST 2025, Problem 3
Determine all positive integers $a, b$ and $c$ such that
$2$ $\cdot$ $10^a + 5^b = 2025^c$
0 replies
Maths_VC
an hour ago
0 replies
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Two permutations
Nima Ahmadi Pour   13
N an hour ago by awesomeming327.
Source: Iran prepration exam
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 + a_2 + \ldots + a_n$.
Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have
\[ n\mid a_i - b_i - c_i \]

Proposed by Ricky Liu & Zuming Feng, USA
13 replies
Nima Ahmadi Pour
Apr 24, 2006
awesomeming327.
an hour ago
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Pls help I'm stuck
UtkarshSri   1
N 2 hours ago by vanstraelen
So I was solving this problem
Cbrt(x+1) - cbrt(x-1) = pow((x²-1),1/6)


Here, the square root is defined in the domain x≤-1 OR x≥1
I substituted a= cbrt(x+1) and b= cbrt(x-1)
And then the equation turned out to be
a-b= sqrt(ab)
Here, since the RHS is +ve, the LHS must be too, so a>b (Done so as to prevent extraneous roots)
Now, I can write the domain in terms of a and b as ab≥0
Squaring, we get a²+b²-3ab=0
On solving we get a/b = (3±sqrt(5))/2
Now, we know that a>b from the above condition, so the only root is a/b= (3+sqrt(5))/2 and on substituting for x, and applying componendo and dividendo after cubing, I get x= sqrt(5)/2

Now, my question is, since the equation is even i.e f(x)=f(-x), it's symmetric about the y-axis, so x= -sqrt(5)/2 must also satisfy the equation, and indeed it does, so where did I made some irreversible step that caused this loss of the root?
when I plugged the equation to wolframalpha, it showed that the "principle root" has the former solⁿ only where's for a "real valued root", it has both.I also found that on solving the equation
b-a= sqrt(ab), I get the latter root.But I mean where did that extra -ve sign come from?
I'm familiar that f(x)=sqrt(g(x)) is equivalent to f²(x)=g(x) AND f(x)≥0 AND g(x)≥0 but the g(x)≥0 condition is not necessary as f²(x) is already ≥0
and also that abs(f(x))=abs(g(x)) is equivalent to f²(x)=g²(x)

PS: Pardon me I don't know latex well, I'm tried to make this as comprehensive as possible and made sure there's no loss of equivalence in any step.
I hope someone clarifies this for me, it's really bothering me for a while.
1 reply
UtkarshSri
Today at 9:57 AM
vanstraelen
2 hours ago
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Calculate the radius of a circle using sidelengths.
richminer   0
3 hours ago
Given triangle ABC with incircle (I), with D being the touchpoint of (I) and BC. Let M be the tangent point of the A-Mixtilinear circle (internally tangent). A' is the reflection of A through I. Calculate the radius of the circle (MDA') using the side lengths of the triangle ABC.
0 replies
richminer
3 hours ago
0 replies
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Number of real roots
girishpimoli   0
4 hours ago
Number of real roots of

$\displaystyle 2\sin(\theta)\cos(3\theta)\sin(5\theta)=-1$
0 replies
girishpimoli
4 hours ago
0 replies
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Practice Problems at chapter end EGMO
idk12345678   4
N Apr 19, 2025 by idk12345678
I am reading the book EGMO rn. I just finished the content and 6 practice problems for chapter 1, and i am wondering what the genrally recommeded amount of problems after each chapter to solve is
4 replies
idk12345678
Apr 18, 2025
idk12345678
Apr 19, 2025
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Practice Problems at chapter end EGMO
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idk12345678
397 posts
idk12345678
#1 h  Apr 18, 2025, 9:26 PM
Y by
I am reading the book EGMO rn. I just finished the content and 6 practice problems for chapter 1, and i am wondering what the genrally recommeded amount of problems after each chapter to solve is
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Pengu14
635 posts
Pengu14
#2 h  Apr 18, 2025, 9:53 PM
Y by
idk12345678 wrote:
I am reading the book EGMO rn. I just finished the content and 6 practice problems for chapter 1, and i am wondering what the genrally recommeded amount of problems after each chapter to solve is

All of them.
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mathprodigy2011
342 posts
mathprodigy2011
#3 h  Apr 18, 2025, 10:36 PM
Y by
Pengu14 wrote:
idk12345678 wrote:
I am reading the book EGMO rn. I just finished the content and 6 practice problems for chapter 1, and i am wondering what the genrally recommeded amount of problems after each chapter to solve is

All of them.

ORZ
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martianrunner
210 posts
martianrunner
#4 h  Apr 19, 2025, 12:20 AM
Y by
poor guy just wanted an answer

i think a fraction greater than or equal to 2/3 of the total practice problems would be good

although maybe in the future you could come back and solve the problems that you left behind
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idk12345678
397 posts
idk12345678
#5 h  Apr 19, 2025, 12:45 AM
Y by
alright thanks. I was thinking of just doing most of them and then coming back later, cuz i was getting rlly bored of angle chasing lol
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