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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

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Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
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0 replies
jlacosta
Mar 2, 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
J
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Flo0r functi0n
m4thbl3nd3r   3
N 12 minutes ago by m4thbl3nd3r
Find all positive integers such that $$n=\lfloor \sqrt{n}\rfloor^2+\lfloor \sqrt{n}\rfloor$$
3 replies
m4thbl3nd3r
42 minutes ago
m4thbl3nd3r
12 minutes ago
J
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Changeable polynomials, can they ever become equal?
mshtand1   4
N 21 minutes ago by CHESSR1DER
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 11.5
Initially, two constant polynomials are written on the board: \(0\) and \(1\). At each step, it is allowed to add \(1\) to one of the polynomials and to multiply another one by the polynomial \(45x + 2025\). Can the polynomials become equal at some point?

Proposed by Oleksii Masalitin
4 replies
mshtand1
Yesterday at 12:47 AM
CHESSR1DER
21 minutes ago
J
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Guessing polynomial by its maximum values on segments
NO_SQUARES   3
N 23 minutes ago by pco
Source: Kvant 2025 no. 1 M2828 and The XIX Southern Mathematical Tournament
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
M. Didin
3 replies
NO_SQUARES
Yesterday at 3:21 PM
pco
23 minutes ago
J
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easy number theory
MuradSafarli   1
N 38 minutes ago by Tuvshuu
\[ v_p(n!) \leq \frac{n}{p - 1} \]
1 reply
MuradSafarli
an hour ago
Tuvshuu
38 minutes ago
J
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Inspired by Kazakhstan 2017
sqing   1
N an hour ago by sqing
Source: Own
Let $a,b,c\ge \frac{1}{2}$ and $a+b+c=2. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 1$$Let $a,b,c\ge \frac{1}{3}$ and $a+b+c=1. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 9$$
1 reply
1 viewing
sqing
2 hours ago
sqing
an hour ago
J
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Collinear geometry problem with incircle
ilovemath0402   1
N an hour ago by deraxenrovalo
Given acute $\triangle ABC$ not isosceles, the incircle $(I)$. $D,E,F$ is the intersection of $(I)$ with $BC,CA,AB$. $P$ is the projection of $D$ onto $EF$. $DP$ cut $(I)$ at the second point $K$. $L$ is the projection of $A$ onto $IK$. $(LEC), (LFB)$ cut $(I)$ at the second point $M,N$ respectively. Prove $M,N,P$ are collinear
1 reply
ilovemath0402
Jul 22, 2023
deraxenrovalo
an hour ago
J
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Wait wasn't it the reciprocal in the paper?
Supercali   6
N 2 hours ago by kes0716
Source: India TST 2023 Day 2 P1
Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$.

Proposed by Navilarekallu Tejaswi
6 replies
Supercali
Jul 9, 2023
kes0716
2 hours ago
J
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About old Inequality
perfect_square   0
2 hours ago
Source: Arqady
This is: $a,b,c>0$ which satisfy $abc=1$
Prove that: $ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$
By $ uvw $ method, I can assum $b=c=x,a=\frac{1}{x^2}$
But I can't prove:
$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $
Is there an another way?
0 replies
perfect_square
2 hours ago
0 replies
J
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inquality
karasuno   1
N 3 hours ago by sqing
The real numbers $x,y,z \ge \frac{1}{2}$ are given such that $x^{2}+y^{2}+z^{2}=1$. Prove the inequality $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2 .$$
1 reply
karasuno
3 hours ago
sqing
3 hours ago
J
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Number Theory
karasuno   0
3 hours ago
Solve the equation $$n!+10^{2014}=m^{4}$$in natural numbers m and n.
0 replies
karasuno
3 hours ago
0 replies
J
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Minimum number of values in the union of sets
bnumbertheory   5
N 3 hours ago by Rohit-2006
Source: Simon Marais Mathematics Competition 2023 Paper A Problem 3
For each positive integer $n$, let $f(n)$ denote the smallest possible value of $$|A_1 \cup A_2 \cup \dots \cup A_n|$$where $A_1, A_2, A_3 \dots A_n$ are sets such that $A_i \not\subseteq A_j$ and $|A_i| \neq |A_j|$ whenever $i \neq j$. Determine $f(n)$ for each positive integer $n$.
5 replies
bnumbertheory
Oct 14, 2023
Rohit-2006
3 hours ago
J
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two triangles have equal circumradii
littletush   5
N 5 hours ago by Taha.kh
Source: Italy TST 2009 p5
Two circles $O_1$ and $O_2$ intersect at $M,N$. The common tangent line nearer to $M$ of the two circles touches $O_1,O_2$ at $A,B$ respectively. Let $C,D$ be the symmetric points of $A,B$ with respect to $M$ respectively. The circumcircle of triangle $DCM$ intersects circles $O_1$ and $O_2$ at points $E,F$ respectively which are distinct from $M$. Prove that the circumradii of the triangles $MEF$ and $NEF$ are equal.
5 replies
littletush
Mar 10, 2012
Taha.kh
5 hours ago
J
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Equal lengths in cyclic quadrilateral
LoloChen   4
N 5 hours ago by Nari_Tom
Source: All-Russian MO 2024 9.4
In cyclic quadrilateral $ABCD$, $\angle A+ \angle D=\frac{\pi}{2}$. $AC$ intersects $BD$ at ${E}$. A line ${l}$ cuts segment $AB, CD, AE, DE$ at $X, Y, Z, T$ respectively. If $AZ=CE$ and $BE=DT$, prove that the diameter of the circumcircle of $\triangle EZT$ equals $XY$.
4 replies
LoloChen
Apr 22, 2024
Nari_Tom
5 hours ago
J
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Two circles and many points
CHN_Lucas   5
N 5 hours ago by Captainscrubz
Source: 2022 China Second Round A2
$A,B,C,D,E$ are points on a circle $\omega$, satisfying $AB=BD$, $BC=CE$. $AC$ meets $BE$ at $P$. $Q$ is on $DE$ such that $BE//AQ$. Suppose $\odot(APQ)$ intersects $\omega$ again at $T$. $A'$ is the reflection of $A$ wrt $BC$. Prove that $A'BPT$ lies on the same circle.
5 replies
CHN_Lucas
Dec 22, 2022
Captainscrubz
5 hours ago
J
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J
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Functional Equation
IstekOlympiadTeam   5
N Feb 10, 2025 by Nuran2010
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\]where $n\in \mathbb{N}.$
5 replies
IstekOlympiadTeam
Feb 19, 2016
Nuran2010
Feb 10, 2025
J
Functional Equation
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Reveal topic tags
function
algebra
functional equation
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IstekOlympiadTeam
542 posts
IstekOlympiadTeam
#1 h  Feb 19, 2016, 2:57 PM • 5 Y
Y by Davi-8191, Amir Hossein, megarnie, Adventure10, Mango247
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\]where $n\in \mathbb{N}.$
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Ankoganit
3070 posts
Ankoganit
#2 h  Feb 19, 2016, 3:01 PM • 3 Y
Y by muradmurad, megarnie, Adventure10
From this post, we conclude there are no solutions.
Also, this is similar to IMO 1987 P4.
This post has been edited 1 time. Last edited by Ankoganit, Feb 19, 2016, 3:04 PM
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Takeya.O
769 posts
Takeya.O
#3 h  Jul 14, 2016, 2:13 PM • 3 Y
Y by megarnie, Adventure10, Mango247
I'm going to write the proof refer to IMO 1987 P4.

Suppose that there is such f.Let A=N－f（N）, B=f（A）.Apparently f is injective.We claim that B=f（N）－f（f（N））.Because if k∈B, k∈f（N） and from injectivity of f, k∈f（N）－f（f（N））.On the other hand, if k∈f（N）－f（f（N））, k∈f（A）=B.

Since A and B are disjoint, |A∪B|=|A|＋|B|.And from injectivity of f, |A|=|B|, so |A∪B| is even.On the other hand, A∪B=N－f（f（N））=｛1,...,2015｝ which is absurd.

Therefore there is no such solution f.
This post has been edited 1 time. Last edited by Takeya.O, Jul 14, 2016, 2:38 PM
Reason: adding explanation
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strong_boy
261 posts
strong_boy
#4 h  Jul 29, 2022, 5:39 PM • 5 Y
Y by hectorleo123, Mango247, Mango247, Mango247, wizixez
It can be easily seen that the function $f$ is injective

Our problem function produces natural values from $2016$ onwards

Suppose the function $f$ alone does not generate $N$ values. So the function $f(f(n))$ does not produce $2N$ value.

So, if a function wants to be true in the problem, the following equation must have a solution


$2N=2015$
And this issue has no natural answer!

So the function does not work in the problem
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MuradSafarli
50 posts
MuradSafarli
#7 h  Feb 8, 2025, 5:54 PM
Y by
Also, this question is the first question in Tutu's book on functions. Book's link-https://www.isinj.com/mt-usamo/Functional%20Equations%20-%20Andresscru,%20Boreico%20(2007).pdf
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Nuran2010
48 posts
Nuran2010
#9 h  Feb 10, 2025, 8:22 PM
Y by
MuradSafarli wrote:
Also, this question is the first question in Tutu's book on functions. Book's link-https://www.isinj.com/mt-usamo/Functional%20Equations%20-%20Andresscru,%20Boreico%20(2007).pdf

Tutu's book on functions
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