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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
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
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China MO 2021 P6
NTssu   22
N 2 minutes ago by HamstPan38825
Source: CMO 2021 P6
Find $f: \mathbb{Z}_+ \rightarrow \mathbb{Z}_+$, such that for any $x,y \in \mathbb{Z}_+$, $$f(f(x)+y)\mid x+f(y).$$
22 replies
NTssu
Nov 25, 2020
HamstPan38825
2 minutes ago
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Very concex function
lomos_lupin   48
N 7 minutes ago by Ilikeminecraft
Source: USAM0 2000 #1 (billzhao)
Call a real-valued function $ f$ very convex if
\[ \frac {f(x) + f(y)}{2} \ge f\left(\frac {x + y}{2}\right) + |x - y| \]
holds for all real numbers $ x$ and $ y$. Prove that no very convex function exists.
48 replies
lomos_lupin
Aug 8, 2005
Ilikeminecraft
7 minutes ago
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Number Theory
MuradSafarli   5
N 19 minutes ago by tchange7575
Find all natural numbers \( a, b, c \) such that

\[ 2^a \cdot 3^b + 1 = 5^c. \]
5 replies
MuradSafarli
2 hours ago
tchange7575
19 minutes ago
J
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USAMO 2001 Problem 6
MithsApprentice   20
N 20 minutes ago by Ritwin
Each point in the plane is assigned a real number such that, for any triangle, the number at the center of its inscribed circle is equal to the arithmetic mean of the three numbers at its vertices. Prove that all points in the plane are assigned the same number.
20 replies
MithsApprentice
Sep 30, 2005
Ritwin
20 minutes ago
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Bijection on the set of integers
talkon   18
N 22 minutes ago by HamstPan38825
Source: InfinityDots MO 2 Problem 2
Determine all bijections $f:\mathbb Z\to\mathbb Z$ satisfying
$$f^{f(m+n)}(mn) = f(m)f(n)$$for all integers $m,n$.

Note: $f^0(n)=n$, and for any positive integer $k$, $f^k(n)$ means $f$ applied $k$ times to $n$, and $f^{-k}(n)$ means $f^{-1}$ applied $k$ times to $n$.

Proposed by talkon
18 replies
+1 w
talkon
Apr 9, 2018
HamstPan38825
22 minutes ago
J
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Functional Equations Marathon March 2025
Levieee   25
N 31 minutes ago by joeym2011
1. before posting another problem please try your best to provide the solution to the previous solution because we don't want a backlog of many problems
2.one is welcome to send functional equations involving calculus (mainly basic real analysis type of proofs) as long it is of the form $\text{"find all functions:"}$
25 replies
+1 w
Levieee
Today at 1:03 AM
joeym2011
31 minutes ago
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Inequality
anhduy98   5
N 41 minutes ago by JK1603JK
Source: Own
Given three real numbers $ a,b,c\ge 0 $ satisfying $: a+b+c=3 $.Prove that:
$$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}+\sqrt{c^2-ca+a^2}\ge 3+\frac{a^2+b^2+c^2-3abc}{3}.$$
5 replies
anhduy98
Oct 28, 2024
JK1603JK
41 minutes ago
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Nice and easy FE on R+
sttsmet   22
N an hour ago by jasperE3
Source: EMC 2024 Problem 4, Seniors
Find all functions $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x+yf(x)) = xf(1+y)$
for all x, y positive reals.
22 replies
sttsmet
Dec 23, 2024
jasperE3
an hour ago
J
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Existence of AP of interesting integers
DVDthe1st   33
N an hour ago by tchange7575
Source: 2018 China TST Day 1 Q2
A number $n$ is interesting if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
33 replies
DVDthe1st
Jan 2, 2018
tchange7575
an hour ago
J
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A diophantine equation
crazyfehmy   13
N an hour ago by Primeniyazidayi
Source: Turkey Junior National Olympiad 2012 P1
Let $x, y$ be integers and $p$ be a prime for which

\[ x^2-3xy+p^2y^2=12p \]
Find all triples $(x,y,p)$.
13 replies
crazyfehmy
Dec 12, 2012
Primeniyazidayi
an hour ago
J
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nf(f(n)) = f(n)^2, f : N->N
Zhero   19
N an hour ago by HamstPan38825
Source: ELMO Shortlist 2010, A1; also ELMO #4
Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$.

Carl Lian and Brian Hamrick.
19 replies
Zhero
Jul 5, 2012
HamstPan38825
an hour ago
J
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RMM 2019 Problem 2
math90   77
N 2 hours ago by ihatemath123
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
77 replies
math90
Feb 23, 2019
ihatemath123
2 hours ago
J
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Two circles concur on a line
math154   59
N 2 hours ago by Mathandski
Source: ELMO Shortlist 2012, G1; also ELMO #1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.

Ray Li.
59 replies
math154
Jul 2, 2012
Mathandski
2 hours ago
J
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functional equation
Anni   7
N 2 hours ago by HamstPan38825
Source: albanian TST 2008 bmo
Find all functions $f: \mathbb R \to \mathbb R$ such that
\[ f(x+f(y))=y+f(x+1),\]for all $x,y \in \mathbb R$.
7 replies
Anni
May 24, 2009
HamstPan38825
2 hours ago
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At least one of 2k-1, 5k-1, 13k-1 is not a perfect square
Demetres   4
N Aug 15, 2022 by the_mathmagician
Source: Cyprus 2022 Junior TST-3 Problem 1
Prove that for every natural number $k$, at least one of the integers
\[ 2k-1, \quad 5k-1 \quad \text{and} \quad 13k-1\]is not a perfect square.
4 replies
Demetres
Mar 26, 2022
the_mathmagician
Aug 15, 2022
J
At least one of 2k-1, 5k-1, 13k-1 is not a perfect square
G H J
Reveal topic tags
number theory
Perfect Square
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Source: Cyprus 2022 Junior TST-3 Problem 1
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Demetres
451 posts
Demetres
#1 h  Mar 26, 2022, 8:29 PM
Y by
Prove that for every natural number $k$, at least one of the integers
\[ 2k-1, \quad 5k-1 \quad \text{and} \quad 13k-1\]is not a perfect square.
Z K Y
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CT17
1481 posts
CT17
#2 h  Mar 26, 2022, 8:47 PM
Y by
If $k$ is even, then $2k-1$ cannot be a perfect square. If $k\equiv 3\pmod{4}$, $k\equiv 5\pmod{16}$, or $k\equiv 9\pmod{16}$, then $5k-1$ cannot be a perfect square. If $k\equiv 1\pmod{16}$ or $k\equiv 13\pmod{16}$, then $13k-1$ cannot be a perfect square. As we have exhausted all residues modulo $16$, we are done.
Z K Y
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sd1006627
107 posts
sd1006627
#3 h  Mar 26, 2022, 9:00 PM
Y by
With process of elimination, if $k$ is $1$, then $2k-1$ would be a perfect square.

If $k$ is $13$, then $5k-1$ can't be a perfect square.

Looking closely at the wording at least one of the integers can't be a perfect square for any natural number $k$.

So, our answer is
$13k-1$
This post has been edited 1 time. Last edited by sd1006627, Mar 26, 2022, 9:00 PM
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MarkBcc168
1593 posts
MarkBcc168
#4 h  Mar 27, 2022, 2:35 AM • 1 Y
Y by guywholovesmathandphysics
IMO 1986/1
Z K Y
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the_mathmagician
467 posts
the_mathmagician
#5 h  Aug 15, 2022, 12:42 AM
Y by
sd1006627 wrote:
With process of elimination, if $k$ is $1$, then $2k-1$ would be a perfect square.

If $k$ is $13$, then $5k-1$ can't be a perfect square.

Looking closely at the wording at least one of the integers can't be a perfect square for any natural number $k$.

So, our answer is
$13k-1$

What this a proof your answer doesn't even make sense

sol
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