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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!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
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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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Combinatorial NT involving sum of fractional parts
Photaesthesia   14
N 32 minutes ago by jjjppp110
Source: 2025 China Mathematical Olympiad Day 1 Problem 3
Let \(a_1, a_2, \ldots, a_n\) be integers such that \(a_1 > a_2 > \cdots > a_n > 1\). Let \(M = \operatorname{lcm} \left( a_1, a_2, \ldots, a_n \right)\). For any finite nonempty set $X$ of positive integers, define \[ f(X) = \min_{1 \leqslant i \leqslant n} \sum_{x \in X} \left\{ \frac{x}{a_i} \right\}. \]Such a set $X$ is called minimal if for every proper subset $Y$ of it, $f(Y) < f(X)$ always holds.

Suppose $X$ is minimal and $f(X) \geqslant \frac{2}{a_n}$. Prove that \[ |X| \leqslant f(X) \cdot M. \]
14 replies
Photaesthesia
Nov 27, 2024
jjjppp110
32 minutes ago
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Abelkonkurransen 2025 4a
Lil_flip38   0
39 minutes ago
Source: abelkonkurransen
Find all polynomials \(P\) with real coefficients satisfying
$$P(\frac{1}{1+x})=\frac{1}{1+P(x)}$$for all real numbers \(x\neq -1\)
0 replies
Lil_flip38
39 minutes ago
0 replies
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p^2+3*p*q+q^2
mathbetter   3
N 40 minutes ago by mathbetter
\[ \text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.} \]
3 replies
mathbetter
Mar 15, 2025
mathbetter
40 minutes ago
J
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Abelkonkurransen 2025 3b
Lil_flip38   0
42 minutes ago
Source: abelkonkurransen
An acute angled triangle \(ABC\) has circumcenter \(O\). The lines \(AO\) and \(BC\) intersect at \(D\), while \(BO\) and \(AC\) intersect at \(E\) and \(CO\) and \(AB\) intersect at \(F\). Show that if the triangles \(ABC\) and \(DEF\) are similar(with vertices in that order), than \(ABC\) is equilateral.
0 replies
Lil_flip38
42 minutes ago
0 replies
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Abelkonkurransen 2025 2b
Lil_flip38   0
an hour ago
Source: abelkonkurransen
Which positive integers $a$ have the property that \(n!-a\) is a perfect square for infinitely many positive integers \(n\)?
0 replies
Lil_flip38
an hour ago
0 replies
J
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Degree inequality
F_Xavier1203   7
N an hour ago by lminsl
Source: 2022 Korea Winter Program Practice Test
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that
$$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$and find when the equality holds.
7 replies
F_Xavier1203
Aug 14, 2022
lminsl
an hour ago
J
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Abelkonkurransen 2025 2a
Lil_flip38   0
an hour ago
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.
0 replies
Lil_flip38
an hour ago
0 replies
J
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Abelkonkurransen 2025 1b
Lil_flip38   0
an hour ago
Source: abelkonkurransen
In Duckville there is a perpetual trophy with the words “Best child of Duckville” engraved on it. Each inhabitant of Duckville has a non-empty list (which never changes) of other inhabitants of Duckville. Whoever receives the trophy
gets to keep it for one day, and then passes it on to someone on their list the next day. Gregers has previously received the trophy. It turns out that each time he does receive it, he is guaranteed to receive it again exactly $2025$ days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers $n>0$ for which we can be absolutely certain that she cannot receive the trophy again in $n$ days, given the above information.
0 replies
Lil_flip38
an hour ago
0 replies
J
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Abelkonkurransen 2025 1a
Lil_flip38   0
an hour ago
Source: abelkonkurransen
Peer and Solveig are playing a game with $n$ coins, all of which show $M$ on one side and $K$ on the opposite side. The coins are laid out in a row on the table. Peer and Solveig alternate taking turns. On his turn, Peer may turn over one or more coins, so long as he does not turn over two adjacent coins. On her turn, Solveig picks precisely two adjacent coins and turns them over. When the game begins, all the coins are showing $M$. Peer plays first, and he wins if all the coins show $K$ simultaneously at any time. Find all $n\geqslant 2$ for which Solveig can keep Peer from winning.
0 replies
Lil_flip38
an hour ago
0 replies
J
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Cool one
MTA_2024   9
N an hour ago by invisibleman
Prove that for all real numbers $a$ and $b$ verifying $a>b>0$ . $$(n+1) \cdot b^n \leq \frac{a^{n+1}-b^{n+1}}{a-b} \leq (n+1) \cdot a^n $$
9 replies
MTA_2024
Mar 15, 2025
invisibleman
an hour ago
J
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Like Father Like Son... (or Like Grandson?)
AlperenINAN   2
N an hour ago by atdaotlohbh
Source: Turkey TST 2025 P4
Let $a,b,c$ be given pairwise coprime positive integers where $a>bc$. Let $m<n$ be positive integers. We call $m$ to be a grandson of $n$ if and only if, for all possible piles of stones whose total mass adds up to $n$ and consist of stones with masses $a,b,c$, it's possible to take some of the stones out from this pile in a way that in the end, we can obtain a new pile of stones with total mass of $m$. Find the greatest possible number that doesn't have any grandsons.
2 replies
AlperenINAN
Mar 18, 2025
atdaotlohbh
an hour ago
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solve the system of equations
Havu   4
N 2 hours ago by LeoaB411
Solve the system of equations:
\[\begin{cases} 3x^2-2xy+3y^2+\dfrac{2}{x^2-2xy+y^2}=8\\ 2x+\dfrac{1}{x-y}=4 \end{cases}\]
4 replies
Havu
Yesterday at 7:52 AM
LeoaB411
2 hours ago
J
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a/b + b/a never integer ?
MTA_2024   4
N 3 hours ago by invisibleman
Let $a$ and $b$ be 2 distinct positive integers.
Can $\frac a b +\frac b a $ be in an integer. Prove why ?
4 replies
MTA_2024
Mar 16, 2025
invisibleman
3 hours ago
J
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Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a p
kyotaro   3
N Today at 4:36 AM by quantam13
Does there exist a function $f:\mathbb N^* \rightarrow \mathbb Z$ other than a polynomial satisfying
$$f(a)-f(b) \mid a-b$$
3 replies
kyotaro
Yesterday at 10:51 AM
quantam13
Today at 4:36 AM
J
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J
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PHP + Fibonacci
SomeonecoolLovesMaths   5
N Mar 16, 2025 by OronSH
The Question

I have tried to search this problem both inside/outside AoPS but was unable to find a solution, specifically with the use of PHP in it. It would be really helpful if someone posted a solution to it.
5 replies
SomeonecoolLovesMaths
Mar 15, 2025
OronSH
Mar 16, 2025
J
PHP + Fibonacci
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Reveal topic tags
Php
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SomeonecoolLovesMaths
3142 posts
SomeonecoolLovesMaths
#1 h  Mar 15, 2025, 10:39 PM
Y by
The Question

I have tried to search this problem both inside/outside AoPS but was unable to find a solution, specifically with the use of PHP in it. It would be really helpful if someone posted a solution to it.
Z K Y
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RoyalPrince
136 posts
RoyalPrince
#2 h  Mar 15, 2025, 10:41 PM
Y by
  There exists a Fibonacci number \(F_{n}\) that ends with 2007 zeros. But I can't find what number it is if that's what you're looking for
Z K Y
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SomeonecoolLovesMaths
3142 posts
SomeonecoolLovesMaths
#3 h  Mar 15, 2025, 10:43 PM
Y by
RoyalPrince wrote:
There exists a Fibonacci number \(F_{n}\) that ends with 2007 zeros. But I can't find what number it is if that's what you're looking for

No not really that. I am looking for a proof for the existence of such a number.
Z K Y
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SomeonecoolLovesMaths
3142 posts
SomeonecoolLovesMaths
#4 h  Mar 15, 2025, 10:59 PM
Y by
I would also love it if you guys could help me with this.

Let $a_1, a_2, \ldots, a_{20}$ be distinct positive integers not exceeding $70$. Show that there is some $k$ so that $a_i - a_j = k$ for four different pairs $(i, j)$.
Z K Y
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mathprodigy2011
226 posts
mathprodigy2011
#5 h  Mar 16, 2025, 1:36 AM
Y by
SomeonecoolLovesMaths wrote:
The Question

I have tried to search this problem both inside/outside AoPS but was unable to find a solution, specifically with the use of PHP in it. It would be really helpful if someone posted a solution to it.

Click to reveal hidden text
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OronSH
1720 posts
OronSH
#6 h  Mar 16, 2025, 2:03 AM
Y by
For any $k$ the sequence $F_n\pmod k$ is periodic, since there are finitely many possible pairs $(F_n,F_{n+1})$ so eventually one will repeat. Additionally since $F_n,F_{n+1}$ uniquely determine $F_{n-1}$ the sequence is purely periodic. Since $F_0=F_2-F_1=0$ there is some other $n$ with $F_n\equiv 0\pmod k$. Then take $k=10^{2007}$
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