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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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Area problem
MTA_2024   1
N 4 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
an hour ago
Curious_Droid
4 minutes ago
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Graph Theory in China TST
steven_zhang123   1
N 7 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
1 viewing
steven_zhang123
4 hours ago
Photaesthesia
7 minutes ago
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Apollonius Circle's Isogonal Conjugate Image
luosw   7
N 15 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
15 minutes ago
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Squares on harmonic quadrilateral
Tkn   1
N 17 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
17 minutes ago
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Expression is a perfect square implies the polynomial is constant
Popescu   3
N 26 minutes ago by luutrongphuc
Source: IMSC Day 2 Problem 3
Let $a\equiv 1\pmod{4}$ be a positive integer. Show that any polynomial $Q\in\mathbb{Z}[X]$ with all positive coefficients such that
$$Q(n+1)((a+1)^{Q(n)}-a^{Q(n)})$$is a perfect square for any $n\in\mathbb{N}^{\ast}$ must be a constant polynomial.

Proposed by Vlad Matei, Romania
3 replies
+1 w
Popescu
Jun 29, 2024
luutrongphuc
26 minutes ago
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Midpoint of bisector; prove that B, E, F, C cyclic
v_Enhance   8
N 26 minutes 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
26 minutes ago
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p^2+3*p*q+q^2
mathbetter   2
N 26 minutes 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
mathbetter
Yesterday at 6:47 PM
togrulhamidli2011
26 minutes ago
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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 an hour 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
an hour ago
J
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Algebra Functions
pear333   1
N an hour 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
an hour ago
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Polynomial equation with rational numbers
Miquel-point   2
N an hour 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
an hour ago
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Geometry
srnjbr   0
2 hours ago
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.
0 replies
srnjbr
2 hours ago
0 replies
J
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Geometry
srnjbr   0
2 hours ago
in triangle abc, l is the leg of bisector a, d is the image of c on line al, and e is the image of l on line ab. take f as the intersection of de and bc. show that af is perpendicular to bc
0 replies
srnjbr
2 hours ago
0 replies
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<QBC =<PCB if BM = CN, <PMC = <MAB, <QNB = < NAC
parmenides51   1
N 2 hours ago by dotscom26
Source: 2005 Estonia IMO Training Test p2
On the side BC of triangle $ABC$, the points $M$ and $N$ are taken such that the point $M$ lies between the points $B$ and $N$, and $| BM | = | CN |$. On segments $AN$ and $AM$, points $P$ and $Q$ are taken so that $\angle PMC = \angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.
1 reply
parmenides51
Sep 24, 2020
dotscom26
2 hours ago
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Inequality over inradius and circumradius of regular n-gon.
Goutham   1
N Sep 16, 2011 by anonymouslonely
Source: ILL 1980-20
The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that
\[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]
1 reply
Goutham
Jun 14, 2011
anonymouslonely
Sep 16, 2011
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Inequality over inradius and circumradius of regular n-gon.
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Reveal topic tags
inequalities
geometry
inradius
circumcircle
geometry unsolved
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Source: ILL 1980-20
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Goutham
3130 posts
Goutham
#1 h  Jun 14, 2011, 2:07 PM • 1 Y
Y by Adventure10
The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that
\[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]
Z K Y
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anonymouslonely
1142 posts
anonymouslonely
#2 h  Sep 16, 2011, 10:25 AM • 2 Y
Y by Adventure10, Mango247
this is equivalent with $ cos(180/n) \geq cos^{2}{180/(n+1)} $.
how to go on?
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