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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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Help meee
hanzo.ei   0
a few seconds ago
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.
0 replies
hanzo.ei
a few seconds ago
0 replies
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D1014 : Intersection of set
Dattier   1
N 5 minutes ago by Dattier
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 $\bigcap \limits_{k=1}^n B_k \neq \emptyset$?
1 reply
Dattier
Yesterday at 12:33 PM
Dattier
5 minutes ago
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a! + b! = 2^{c!}
parmenides51   5
N 10 minutes ago by TigerOnion
Source: 2023 Austrian Mathematical Olympiad, Junior Regional Competition , Problem 4
Determine all triples $(a, b, c)$ of positive integers such that
$$a! + b! = 2^{c!}.$$
(Walther Janous)
5 replies
parmenides51
Mar 26, 2024
TigerOnion
10 minutes ago
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Geometry
srnjbr   1
N 17 minutes ago by Curious_Droid
In triangle ABC, D is the leg of the altitude from A. l is a variable line passing through D. E and F are points on l such that AEB=AFC=90. Find the locus of the midpoint of the line segment EF.
1 reply
srnjbr
2 hours ago
Curious_Droid
17 minutes ago
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Area problem
MTA_2024   1
N 24 minutes ago by Curious_Droid
Let $\omega$ be a circle inscribed inside a rhombus $ABCD$. Let $P$ and $Q$ be variable points on $AB$ and $AD$ respectively, such as $PQ$ is always the tangent line to $\omega$.
Prove that for any position of $P$ and $Q$ the area of triangle $\triangle CPQ$ is the same.
1 reply
MTA_2024
2 hours ago
Curious_Droid
24 minutes ago
J
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Graph Theory in China TST
steven_zhang123   1
N 27 minutes ago by Photaesthesia
Source: China TST 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).
1 reply
steven_zhang123
4 hours ago
Photaesthesia
27 minutes ago
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Apollonius Circle's Isogonal Conjugate Image
luosw   7
N 35 minutes ago by drmzjoseph
How to prove: the isogonal image of the Apollonius circle of $\triangle ABC$ passing through point $A$ is an oblique strophoid.

IMAGE
7 replies
luosw
Jun 6, 2021
drmzjoseph
35 minutes ago
J
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Squares on harmonic quadrilateral
Tkn   1
N 37 minutes ago by EeEeRUT
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 $\overleftrightarrow{OX}$.
1 reply
Tkn
5 hours ago
EeEeRUT
37 minutes ago
J
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Midpoint of bisector; prove that B, E, F, C cyclic
v_Enhance   8
N an hour ago by ihategeo_1969
Source: Taiwan 2014 TST1, Problem 4
Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.
8 replies
v_Enhance
Jul 18, 2014
ihategeo_1969
an hour ago
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p^2+3*p*q+q^2
mathbetter   2
N an hour ago by togrulhamidli2011
\[ \text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.} \]
2 replies
1 viewing
mathbetter
Yesterday at 6:47 PM
togrulhamidli2011
an hour ago
J
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P(a+1)=1 for every root a
MTA_2024   1
N an hour ago by pco
Find all real polynomials $P$ of degree $K$ having $k$ distinct real roots ($k \in \mathbb N$), such that for every root $a$ : $P(a+1)=1$.
1 reply
MTA_2024
an hour ago
pco
an hour ago
J
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Functional equation
socrates   30
N 2 hours ago by NicoN9
Source: Baltic Way 2014, Problem 4
Find all functions $f$ defined on all real numbers and taking real values such that \[f(f(y)) + f(x - y) = f(xf(y) - x),\] for all real numbers $x, y.$
30 replies
socrates
Nov 11, 2014
NicoN9
2 hours ago
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Algebra Functions
pear333   1
N 2 hours ago by whwlqkd
Let $P(z)=z-1/z$. Prove that there does not exist a pair of rational numbers $x,y$ such that $P(x)+P(y)=4$.
1 reply
pear333
Today at 12:20 AM
whwlqkd
2 hours ago
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Polynomial equation with rational numbers
Miquel-point   2
N 2 hours ago by Assassino9931
Source: Romanian TST 1979 day 2 P1
Determine the polynomial $P\in \mathbb{R}[x]$ for which there exists $n\in \mathbb{Z}_{>0}$ such that for all $x\in \mathbb{Q}$ we have: \[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]
Dumitru Bușneag
2 replies
Miquel-point
Apr 15, 2023
Assassino9931
2 hours ago
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Problem 3 of First round
Pinko   2
N Nov 27, 2019 by sqing
Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade
If $x$ and $y$ are real numbers, determine the greatest possible value of the expression
$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.
2 replies
Pinko
Nov 27, 2019
sqing
Nov 27, 2019
J
Problem 3 of First round
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Reveal topic tags
algebra
maximum value
inequalities
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G H BBookmark kLocked kLocked NReply
Source: II International Festival of Young Mathematicians Sozopol 2011, Theme for 10-12 grade
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Pinko
437 posts
Pinko
#1 h  Nov 27, 2019, 10:19 AM • 1 Y
Y by Adventure10
If $x$ and $y$ are real numbers, determine the greatest possible value of the expression
$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.
Z K Y
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bel.jad5
3750 posts
bel.jad5
#2 h  Nov 27, 2019, 11:10 AM • 2 Y
Y by Pinko, Adventure10
Pinko wrote:
If $x$ and $y$ are real numbers, determine the greatest possible value of the expression
$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)}$.
We have:
$$(x^2+1)^2(y^2+1)^2=(x^2+1)(y^2+1)(x^2+1)(y^2+1) \geq \frac{(x+1)^2(y+1)^2(xy+1)^2}{4}$$Therefore:
$$\frac{(x+1)(y+1)(xy+1)}{(x^2+1)(y^2+1)} \leq 2$$Equality at $x=y=1$
This post has been edited 1 time. Last edited by bel.jad5, Nov 27, 2019, 11:24 AM
Z K Y
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sqing
41020 posts
sqing
#3 h  Nov 27, 2019, 1:47 PM • 2 Y
Y by Adventure10, Mango247
If $x$ and $y$ are real numbers, determine the greatest possible value of the expression
$\frac{(x+1)(y+1)(3xy+1)}{(x^2+1)(y^2+1)}.$
Z K Y
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