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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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people in the circle
Pomegranat   0
44 minutes ago
Source: idk

Let $n \geq 5$ people be arranged in a circle, numbered clockwise from $1$ to $n$. These people are eliminated one by one in order, until only one person remains. The elimination follows this rule: among the remaining people, start counting clockwise from the person with the smallest number, and eliminate the $n$-th person in that count. Then, among the remaining people, start counting again from the person with the smallest number and eliminate the $n$-th person. Repeat this process until only one person remains. Let $W(n)$ denote the number of the last remaining person.

For example, when $n = 5$, people are eliminated in the following order: $5, 1, 3, 2$. Thus, $W(5) = 4$. It is known that $W(n) = n - 4$ under certain conditions. Prove that the necessary and sufficient condition for this is that both $n + 1$ and $n/2$ are prime numbers.
0 replies
Pomegranat
44 minutes ago
0 replies
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ISI UGB 2025 P4
SomeonecoolLovesMaths   5
N an hour ago by mqoi_KOLA
Source: ISI UGB 2025 P4
Let $S^1 = \{ z \in \mathbb{C} \mid |z| =1 \}$ be the unit circle in the complex plane. Let $f \colon S^1 \longrightarrow S^2$ be the map given by $f(z) = z^2$. We define $f^{(1)} \colon = f$ and $f^{(k+1)} \colon = f \circ f^{(k)}$ for $k \geq 1$. The smallest positive integer $n$ such that $f^{(n)}(z) = z$ is called the period of $z$. Determine the total number of points in $S^1$ of period $2025$.
(Hint : $2025 = 3^4 \times 5^2$)
5 replies
SomeonecoolLovesMaths
Yesterday at 11:24 AM
mqoi_KOLA
an hour ago
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hard inequality omg
tokitaohma   3
N an hour ago by tokitaohma
1. Given $a, b, c > 0$ and $abc=1$
Prove that: $ \sqrt{a^2+1} + \sqrt{b^2+1} + \sqrt{c^2+1} \leq \sqrt{2}(a+b+c) $

2. Given $a, b, c > 0$ and $a+b+c=1 $
Prove that: $ \dfrac{\sqrt{a^2+2ab}}{\sqrt{b^2+2c^2}} + \dfrac{\sqrt{b^2+2bc}}{\sqrt{c^2+2a^2}} + \dfrac{\sqrt{c^2+2ca}}{\sqrt{a^2+2b^2}} \geq \dfrac{1}{a^2+b^2+c^2} $
3 replies
tokitaohma
Yesterday at 5:24 PM
tokitaohma
an hour ago
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Divisibilty...
Sadigly   5
N an hour ago by COCBSGGCTG3
Source: Azerbaijan Junior NMO 2025 P2
Find all $4$ consecutive even numbers, such that the sum of their squares divides the square of their product.
5 replies
Sadigly
Saturday at 9:07 PM
COCBSGGCTG3
an hour ago
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Diophantine involving cube
Sadigly   1
N 2 hours ago by mashumaro
Source: Azerbaijan Senior NMO 2020
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$
1 reply
Sadigly
6 hours ago
mashumaro
2 hours ago
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Old hard problem
ItzsleepyXD   2
N 2 hours ago by ItzsleepyXD
Source: IDK
Let $ABC$ be a triangle and let $O$ be its circumcenter and $I$ its incenter.
Let $P$ be the radical center of its three mixtilinears and let $Q$ be the isogonal conjugate of $P$.
Let $G$ be the Gergonne point of the triangle $ABC$.
Prove that line $QG$ is parallel with line $OI$ .
2 replies
ItzsleepyXD
Apr 25, 2025
ItzsleepyXD
2 hours ago
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The Return of Triangle Geometry
peace09   9
N 3 hours ago by mathfun07
Source: 2023 ISL A7
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]for every $k=1,2,\dots,N$.
9 replies
peace09
Jul 17, 2024
mathfun07
3 hours ago
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Set Partition
Butterfly   0
3 hours ago
For the set of positive integers $\{1,2,…,n\}(n\ge 3)$, no matter how its elements are partitioned into two subsets, at least one of the subsets must contain three numbers $a,b,c$ ($a=b$ is allowed) such that $ab=c$. Find the minimal $n$.
0 replies
Butterfly
3 hours ago
0 replies
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Points Lying on its Cevian Triangle's Thomson Cubic
Feuerbach-Gergonne   1
N 3 hours ago by golue3120
Source: Own
Given $\triangle ABC$ and a point $P$, let $\triangle DEF$ be the cevian triangle of $P$ with respect to $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$, and denote the isotomic conjugate of $H, P$ with respect to $\triangle ABC$ by $X, Q$, respectively. Let the centroid of $\triangle DEF$ be $M$, and denote the isogonal conjugate of $P$ with respect to $\triangle DEF$ by $R$. Prove that
$$ P, Q, X \text{ are collinear} \iff P, R, M \text{ are collinear}. $$or in brief
$$ P \in \text{ K007 of } \triangle ABC \iff P \in \text{ K002 of } \triangle DEF. $$
1 reply
Feuerbach-Gergonne
Jul 19, 2024
golue3120
3 hours ago
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Areas of triangles AOH, BOH, COH
Arne   71
N 3 hours ago by EpicBird08
Source: APMO 2004, Problem 2
Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Prove that the area of one of the triangles $AOH$, $BOH$ and $COH$ is equal to the sum of the areas of the other two.
71 replies
Arne
Mar 23, 2004
EpicBird08
3 hours ago
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Problem 6
termas   68
N 4 hours ago by HamstPan38825
Source: IMO 2016
There are $n\ge 2$ line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands $n-1$ times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.

(a) Prove that Geoff can always fulfill his wish if $n$ is odd.

(b) Prove that Geoff can never fulfill his wish if $n$ is even.
68 replies
1 viewing
termas
Jul 12, 2016
HamstPan38825
4 hours ago
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2n^2+4n-1 and 3n+4 have common powers
bin_sherlo   2
N 4 hours ago by Assassino9931
Source: Türkiye 2025 JBMO TST P5
Find all positive integers $n$ such that a positive integer power of $2n^2+4n-1$ equals to a positive integer power of $3n+4$.
2 replies
bin_sherlo
Yesterday at 7:13 PM
Assassino9931
4 hours ago
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combi/nt
blug   2
N 4 hours ago by aaravdodhia
Prove that every positive integer $n$ can be written in the form
$$n=a_1+a_2+...+a_k,$$where $a_m=2^i3^j$ for some non-negative $i, j$ such that
$$a_x\nmid a_y$$for every $x, y\leq k$.
2 replies
1 viewing
blug
May 9, 2025
aaravdodhia
4 hours ago
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System of equations in juniors' exam
AlperenINAN   2
N 4 hours ago by Assassino9931
Source: Turkey JBMO TST 2025 P3
Find all positive real solutions $(a, b, c)$ to the following system:
$$ \begin{aligned} a^2 + \frac{b}{a} &= 8, \\ ab + c^2 &= 18, \\ 3a + b + c &= 9\sqrt{3}. \end{aligned} $$
2 replies
AlperenINAN
Yesterday at 7:41 PM
Assassino9931
4 hours ago
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J
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MM 2201 (Symmetric Inequality with Weird Sharp Case)
kgator   1
N Apr 10, 2025 by CHESSR1DER
Source: Mathematics Magazine Volume 97 (2024), Issue 4: https://doi.org/10.1080/0025570X.2024.2393998
2201. Proposed by Leonard Giugiuc, Drobeta-Turnu Severin, Romania. Find all real numbers $K$ such that
$$a^2 + b^2 + c^2 - 3 \geq K(a + b + c - 3)$$for all nonnegative real numbers $a$, $b$, and $c$ with $abc \leq 1$.
1 reply
kgator
Apr 10, 2025
CHESSR1DER
Apr 10, 2025
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MM 2201 (Symmetric Inequality with Weird Sharp Case)
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Reveal topic tags
inequalities
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Source: Mathematics Magazine Volume 97 (2024), Issue 4: https://doi.org/10.1080/0025570X.2024.2393998
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kgator
120 posts
kgator
#1 h  Apr 10, 2025, 6:19 PM
Y by
2201. Proposed by Leonard Giugiuc, Drobeta-Turnu Severin, Romania. Find all real numbers $K$ such that
$$a^2 + b^2 + c^2 - 3 \geq K(a + b + c - 3)$$for all nonnegative real numbers $a$, $b$, and $c$ with $abc \leq 1$.
Z K Y
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CHESSR1DER
58 posts
CHESSR1DER
#2 h  Apr 10, 2025, 7:11 PM
Y by
Taking $(e,1,\frac{1}{e})$ we get $e^4-2e^2+1\geq e(e^2-2e+1)K$.
So $K \leq e-2+ \frac{1}{e}$.
Since we can choose $e$ infinitely close to 1, we have $K \leq 2$.
Taking $(1-e,1-e,1-e)$ we get $3e^2-6e \geq -3eK$
So $K \geq 2-e$.
Since we can choose $e$ infinitely small, we have $K \geq 2$.
If $K = 2$ we have $a^2+b^2+c^2-3 \geq (2a-1)+(2b-1)+(2c-1)-3 = 2(a+b+c-3)$.
Answer: $K=2$.
This post has been edited 2 times. Last edited by CHESSR1DER, Apr 10, 2025, 7:46 PM
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