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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.
[*]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
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
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
Perfect Squares, Infinite Integers and Integers
steven_zhang123   0
12 minutes ago
Source: China TST 2001 Quiz 5 P1
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
0 replies
steven_zhang123
12 minutes ago
0 replies
f(f(x)+y)+f(x+y)=2x+2f(y)
parmenides51   3
N 14 minutes ago by Burmf
Source: 2015 AGCN Competition p1 by bobthesmartypants https://artofproblemsolving.com/community/c5h1128876p5232794
Find all functions $f:\mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0}$ satisfying$$f(f(x)+y)+f(x+y)=2x+2f(y)$$
3 replies
parmenides51
Dec 5, 2023
Burmf
14 minutes ago
Help to prove an inequality
JK1603JK   1
N 17 minutes ago by jawadkaleem
Source: unknown
If a,b,c\ge 0: ab+bc+ca=1 then prove \frac{a\left(b+c+2\right)}{bc+2a}+\frac{b\left(c+a+2\right)}{ca+2b}+\frac{c\left(a+b+2\right)}{ab+2c}\ge 3
* Please help me convert it to latex form. Thank you.
1 reply
JK1603JK
32 minutes ago
jawadkaleem
17 minutes ago
2^a + 3^b + 5^c = n!
togrulhamidli2011   2
N 20 minutes ago by togrulhamidli2011
\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]\[
2^a + 3^b + 5^c = n!
\]
2 replies
togrulhamidli2011
30 minutes ago
togrulhamidli2011
20 minutes ago
[ELMO1] System of Functional Equations
v_Enhance   27
N 37 minutes ago by NicoN9
Source: ELMO 2014, Problem 1, by Evan Chen
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
  f(x+f(y)) &= g(x) + h(y) \\
  g(x+g(y)) &= h(x) + f(y) \\
  h(x+h(y)) &= f(x) + g(y) 
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is injective if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)

Proposed by Evan Chen
27 replies
v_Enhance
Jun 30, 2014
NicoN9
37 minutes ago
Graph Theory in China TST
steven_zhang123   2
N an hour ago by steven_zhang123
Source: China TST 2001 Quiz 4 P3
For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).
2 replies
steven_zhang123
Today at 5:42 AM
steven_zhang123
an hour ago
2015 Azerbaijan IMO TST
IstekOlympiadTeam   3
N an hour ago by MuradSafarli
Source: 2015 Azerbaijan IMO TST
We say that $A$$=${$a_1,a_2,a_3\cdots a_n$} consisting $n>2$ distinct positive integers is $good$ if for every $i=1,2,3\cdots n$ the number ${a_i}^{2015}$ is divisible by the product of all numbers in $A$ except $a_i$. Find all integers $n>2$ such that exists a $good$ set consisting of $n$ positive integers.
3 replies
IstekOlympiadTeam
May 29, 2015
MuradSafarli
an hour ago
D1014 : Intersection of set
Dattier   2
N an hour ago by CatinoBarbaraCombinatoric
Source: les dattes à Dattier
Let $A=\{1,...,n\}$ with $\forall i\in A,B_i \subset A$ and $\forall i \in A, \text{card}(\bigcap \limits_{k=1,k\neq i}^n B_k)\geq 2$.

Is it true that $\bigcap \limits_{k=1}^n B_k \neq \emptyset$?
2 replies
Dattier
Yesterday at 12:33 PM
CatinoBarbaraCombinatoric
an hour ago
Inequality => square
Rushil   10
N an hour ago by mqoi_KOLA
Source: INMO 1998 Problem 4
Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.
10 replies
Rushil
Oct 7, 2005
mqoi_KOLA
an hour ago
postaffteff
JetFire008   5
N 2 hours ago by drmzjoseph
Source: Internet
Let $P$ be the Fermat point of a $\triangle ABC$. Prove that the Euler line of the triangles $PAB$, $PBC$, $PCA$ are concurrent and the point of concurrence is $G$, the centroid of $\triangle ABC$.
5 replies
JetFire008
Yesterday at 12:33 PM
drmzjoseph
2 hours ago
inequality marathon
EthanWYX2009   189
N 2 hours ago by Quantum-Phantom
There is an inequality marathon now, but the problem is too hard for me to solve, let's start a new one here, please post problems that is not too difficult.
------
P1.
Find the maximum value of ${M}$, such that for $\forall a,b,c\in\mathbb R_+,$
$$a^3+b^3+c^3-3abc\geqslant M(a^2b+b^2c+c^2a-3abc).$$
189 replies
1 viewing
EthanWYX2009
May 21, 2023
Quantum-Phantom
2 hours ago
Help meee
hanzo.ei   1
N 2 hours ago by Curious_Droid
Given a triangle ABC with (O) and (I) as its circumcircle and incircle, respectively. The incircle (I) touches BC, CA, AB at D, E, F, respectively. Prove that OI passes through the centroid of triangle DEF.
1 reply
hanzo.ei
2 hours ago
Curious_Droid
2 hours ago
D1010 : How it is possible ?
Dattier   10
N 2 hours ago by Dattier
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
10 replies
Dattier
Mar 10, 2025
Dattier
2 hours ago
Squares on harmonic quadrilateral
Tkn   2
N 2 hours ago by Curious_Droid
Let $ABCD$ be a cyclic quadrilateral for which $AB\cdot CD=AD\cdot BC$. Construct two squares $BCEF$ and $DCGH$ externally on the sides $\overline{BC}$ and $\overline{DC}$ respectively.
Suppose that $\overleftrightarrow{BD}$ meets $\overleftrightarrow{AC}$ at $X$, $\overleftrightarrow{BE}$ meets $\overleftrightarrow{DG}$ at $Z$ and $O$ denotes circumcenter of $ABCD$. Prove that $(ZEG)$ and $(ZBD)$ meets again on $\overleftrightarrow{OX}$.
2 replies
Tkn
Today at 5:10 AM
Curious_Droid
2 hours ago
111's in this strange sequence?
ZeusDM   2
N Nov 18, 2019 by Letteer
Source: Rio de Janeiro Mathematical Olympiad 2018, Level 4, #2
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?
2 replies
ZeusDM
Nov 17, 2019
Letteer
Nov 18, 2019
111's in this strange sequence?
G H J
G H BBookmark kLocked kLocked NReply
Source: Rio de Janeiro Mathematical Olympiad 2018, Level 4, #2
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ZeusDM
102 posts
#1 • 2 Y
Y by Adventure10, Mango247
Let $(a_n)$ be a sequence of integers, with $a_1 = 1$ and for evert integer $n \ge 1$, $a_{2n} = a_n + 1$ and $a_{2n+1} = 10a_n$. How many times $111$ appears on this sequence?
Z K Y
The post below has been deleted. Click to close.
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rchokler
2930 posts
#2 • 1 Y
Y by Adventure10
If no $\times10$ steps, we get $1$ routes.

If $1$ step of $\times 10$ then there are $11$ routes since this step happens before you exceed $11$.

If $2$ step of $\times 10$ then there are $2$ routes since the first such step must happen immediately.

There are no routes involving more than $2$ steps of $\times 10$.

That is a total of $14$ routes. This is also how many times $111$ happens since every positive integer is uniquely reachable from $1$ by a sequence of $n\mapsto 2n$ and $n\mapsto 2n+1$ moves.

The indices where $111$ happens are:

$14$
$4098$
$14336$
$2099200$
$1075838976$
$551903297536$
$283673999966208$
$146366987889541120$
$76092819304051900416$
$40140115104391984316416$
$21760664753063325144711168$
$12379400392853802748991242240$
$7605903601369376408980219232256$
$1298074214633706907132624082305024$
Z K Y
The post below has been deleted. Click to close.
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Letteer
56 posts
#3 • 1 Y
Y by Adventure10
Solution
Z K Y
N Quick Reply
G
H
=
a