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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.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

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.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
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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
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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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Inequality by Po-Ru Loh
v_Enhance   54
N 3 minutes ago by Marcus_Zhang
Source: ELMO 2003 Problem 4
Let $x,y,z \ge 1$ be real numbers such that \[ \frac{1}{x^2-1} + \frac{1}{y^2-1} + \frac{1}{z^2-1} = 1. \] Prove that \[ \frac{1}{x+1} + \frac{1}{y+1} + \frac{1}{z+1} \le 1. \]
54 replies
+1 w
v_Enhance
Dec 29, 2012
Marcus_Zhang
3 minutes ago
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Problem 5
Functional_equation   14
N 15 minutes ago by ali123456
Source: Azerbaijan third round 2020(JBMO Shortlist 2019 N6)
$a,b,c$ are non-negative integers.
Solve: $a!+5^b=7^c$

Proposed by Serbia
14 replies
Functional_equation
Jun 6, 2020
ali123456
15 minutes ago
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a^12+3^b=1788^c
falantrng   6
N an hour ago by ali123456
Source: Azerbaijan NMO 2024. Junior P3
Find all the natural numbers $a, b, c$ satisfying the following equation:
$$a^{12} + 3^b = 1788^c$$.
6 replies
falantrng
Jul 8, 2024
ali123456
an hour ago
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stuck on a system of recurrence sequence
Nonecludiangeofan   0
an hour ago
Please guys help me solve this nasty problem that i've been stuck for the past month:
Let \( (a_n) \) and \( (b_n) \) be two sequences defined by:
\[ a_{n+1} = \frac{1 + a_n + a_n b_n}{b_n} \quad \text{and} \quad b_{n+1} = \frac{1 + b_n + a_n b_n}{a_n} \]for all \( n \ge 0 \), with initial values \( a_0 = 1 \) and \( b_0 = 2 \).

Prove that:
\[ a_{2024} < 5. \]
(btw am still not comfortable with system of recurrence sequences)
0 replies
Nonecludiangeofan
an hour ago
0 replies
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A huge group of children compare their heights
Tintarn   5
N an hour ago by InCtrl
Source: All-Russian MO 2024 9.8
$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.)
Proposed by I. Bogdanov
5 replies
Tintarn
Apr 22, 2024
InCtrl
an hour ago
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Iran Inequality
mathmatecS   15
N 2 hours ago by Marcus_Zhang
Source: Iran 1998
When $x(\ge1),$ $y(\ge1),$ $z(\ge1)$ satisfy $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2,$ prove in equality.
$$\sqrt{x+y+z}\ge\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}$$
15 replies
mathmatecS
Jun 11, 2015
Marcus_Zhang
2 hours ago
J
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Inequality involving x, y and z
cefer   46
N 2 hours ago by Baimukh
Source: Balkan MO 2012 - Problem 2
Prove that
\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]
for all positive real numbers $x,y$ and $z$.
46 replies
cefer
Apr 28, 2012
Baimukh
2 hours ago
J
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Numbers from 1 to 15 with rare properties
hectorleo123   1
N 2 hours ago by EmersonSoriano
Source: 2015 Peru Cono Sur TST P2
Let $a, b, c$ and $d$ be elements of the set $\{ 1, 2, 3,\ldots , 2014, 2015 \}$ such that $a < b < c < d$, $a + b$ is a divisor of $c + d$, and $a + c$ is a divisor of $b + d$. Determine the largest value that $a$ can take.
1 reply
hectorleo123
Jul 10, 2023
EmersonSoriano
2 hours ago
J
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Number Theory
MuradSafarli   3
N 3 hours ago by MuradSafarli
find all natural numbers \( (a, b) \) such that the following equation holds:

\[ 7^a + 1 = 2b^2 \]
3 replies
MuradSafarli
4 hours ago
MuradSafarli
3 hours ago
J
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Abelkonkurransen 2025 2a
Lil_flip38   1
N 4 hours ago by RANDOM__USER
Source: Abelkonkurransen
A teacher asks each of eleven pupils to write a positive integer with at most four digits, each on a separate yellow sticky note. Show that if all the numbers are different, the teacher can always submit two or more of the eleven stickers so that the average of the numbers on the selected notes are not an integer.
1 reply
Lil_flip38
Today at 11:10 AM
RANDOM__USER
4 hours ago
J
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Abelkonkurransen 2025 2b
Lil_flip38   3
N 4 hours ago by alexanderhamilton124
Source: abelkonkurransen
Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?
3 replies
Lil_flip38
Today at 11:12 AM
alexanderhamilton124
4 hours ago
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Oi! These lines concur
Rg230403   18
N 5 hours ago by HoRI_DA_GRe8
Source: LMAO 2021 P5, LMAOSL G3(simplified)
Let $I, O$ and $\Gamma$ respectively be the incentre, circumcentre and circumcircle of triangle $ABC$. Points $A_1, A_2$ are chosen on $\Gamma$, such that $AA_1 = AI = AA_2$, and point $A'$ is the foot of the altitude from $I$ to $A_1A_2$. If $B', C'$ are similarly defined, prove that lines $AA', BB'$ and $CC'$ concurr on $OI$.
Original Version from SL
Proposed by Mahavir Gandhi
18 replies
Rg230403
May 10, 2021
HoRI_DA_GRe8
5 hours ago
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Incircle
PDHT   0
6 hours ago
Source: Nguyen Minh Ha
Given a triangle \(ABC\) that is not isosceles at \(A\), let \((I)\) be its incircle, which is tangent to \(BC, CA, AB\) at \(D, E, F\), respectively. The lines \(DE\) and \(DF\) intersect the line passing through \(A\) and parallel to \(BC\) at \(M\) and \(N\), respectively. The lines passing through \(M, N\) and perpendicular to \(MN\) intersect \(IF\) and \(IE\) at \(Q\) and \(P\), respectively.

Prove that \(D, P, Q\) are collinear and that \(PF, QE, DI\) are concurrent.
0 replies
PDHT
6 hours ago
0 replies
J
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Surprisingly low answer to the question what is the maximum
mshtand1   2
N 6 hours ago by sarjinius
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 8.6, 10.5
Given $2025$ positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining $2024$ numbers that are not hidden. What is the maximum number of the $2025$ written numbers that can be perfect squares?

Proposed by Oleksii Masalitin
2 replies
mshtand1
Mar 13, 2025
sarjinius
6 hours ago
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J
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classical R+ FE
jasperE3   2
N Yesterday at 6:17 PM by jasperE3
Source: kent2207, based on 2019 Slovenia TST
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions $f:\mathbb R^+\to\mathbb R^+$ for which:
$$f(f(x)+f(y))=yf(1+yf(x))$$for all $x,y\in\mathbb R^+$.
2 replies
jasperE3
Mar 18, 2025
jasperE3
Yesterday at 6:17 PM
J
classical R+ FE
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Reveal topic tags
functional equation
fe
algebra
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Source: kent2207, based on 2019 Slovenia TST
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jasperE3
11100 posts
jasperE3
#1 h  Mar 18, 2025, 3:55 PM
Y by
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions $f:\mathbb R^+\to\mathbb R^+$ for which:
$$f(f(x)+f(y))=yf(1+yf(x))$$for all $x,y\in\mathbb R^+$.
Z K Y
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pco
23450 posts
pco
#2 h  Yesterday at 10:22 AM • 1 Y
Y by jasperE3
jasperE3 wrote:
wanted to post this problem in its own thread: https://artofproblemsolving.com/community/c6h1784825p34307772
Find all functions $f:\mathbb R^+\to\mathbb R^+$ for which:
$$f(f(x)+f(y))=yf(1+yf(x))$$for all $x,y\in\mathbb R^+$.
Let $P(x,y)$ be the assertion $f(f(x)+f(y))=yf(1+yf(x))$

$P(x+1,\frac x{f(x+1)})$ $\implies$ $f(f(x+1)+f(\frac x{f(x+1)}))=x$ and so $f(x)$ is surjective

So $P(x,y)$ be be written as new assertion $Q(x,y)$ : $f(x+f(y))=yf(1+xy)$

1) $f(x)\le\frac 1x$ $\forall x>0$
Proof

2) $f(x)$ is injective
Proof

3) $\boxed{f(x)=\frac 1x\quad\forall x>0}$
Proof
Z K Y
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jasperE3
11100 posts
jasperE3
#3 h  Yesterday at 6:17 PM
Y by
My solution
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