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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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Geometry
srnjbr   0
10 minutes 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
10 minutes ago
0 replies
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<QBC =<PCB if BM = CN, <PMC = <MAB, <QNB = < NAC
parmenides51   1
N 25 minutes 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
25 minutes ago
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Bosnia and Herzegovina EGMO TST 2017 Problem 2
gobathegreat   2
N 37 minutes ago by anvarbek0813
Source: Bosnia and Herzegovina EGMO Team Selection Test 2017
It is given triangle $ABC$ and points $P$ and $Q$ on sides $AB$ and $AC$, respectively, such that $PQ\mid\mid BC$. Let $X$ and $Y$ be intersection points of lines $BQ$ and $CP$ with circumcircle $k$ of triangle $APQ$, and $D$ and $E$ intersection points of lines $AX$ and $AY$ with side $BC$. If $2\cdot DE=BC$, prove that circle $k$ contains intersection point of angle bisector of $\angle BAC$ with $BC$
2 replies
gobathegreat
Sep 19, 2018
anvarbek0813
37 minutes ago
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Another NT FE
nukelauncher   58
N an hour ago by andrewthenerd
Source: ISL 2019 N4
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
58 replies
nukelauncher
Sep 22, 2020
andrewthenerd
an hour ago
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Easiest Functional Equation
NCbutAN   7
N an hour ago by InftyByond
Source: Random book
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(yf(x)+f(xy))=(x+f(x))f(y)$$Follows for all reals $x,y$.
7 replies
NCbutAN
Mar 2, 2025
InftyByond
an hour ago
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Lower bound for integer relatively prime to n
62861   23
N 2 hours ago by YaoAOPS
Source: USA Winter Team Selection Test #1 for IMO 2018, Problem 1
Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds.

Proposed by Ashwin Sah
23 replies
62861
Dec 11, 2017
YaoAOPS
2 hours ago
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Number of quadratic residues mod $p$ up to $x$
Gauler   0
2 hours ago
Call an integer $n$ to be $y$-smooth if all of its prime factors are $\le y$. Let $\Psi(x,y)$ denote the number of $y$-smooth integers upto $x$. Let $y = y(p)$ be the least quadratic non-residue mod $p$. Show that there are at least $\Psi(x,y)$ quadratic residues mod $p$ up to $x$.
0 replies
Gauler
2 hours ago
0 replies
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Property of Arithmetic function
Gauler   0
2 hours ago
If $f$ is a non-negative arithmetic function and
$$F(\sigma)=\sum_{n=1}^\infty \frac{f(n)}{n^\sigma}$$is convergent for some $0<\sigma<1$, then prove that
$$\sum_{n\le x} f(n) + x\sum_{n>x} \frac{f(n)}{n} \le x^\sigma F(\sigma).$$
0 replies
Gauler
2 hours ago
0 replies
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Graph Theory in China TST
steven_zhang123   0
2 hours ago
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).
0 replies
steven_zhang123
2 hours ago
0 replies
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Directed Paths and Cycles
steven_zhang123   0
2 hours ago
Source: China TST Quiz 4 P2.2
Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).
0 replies
steven_zhang123
2 hours ago
0 replies
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Counting Circles and Maximizing Edges
steven_zhang123   0
3 hours ago
Source: China TST 2001 Quiz 4 P2.1
Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \).

Answer the following questions:
1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph.
2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.
0 replies
steven_zhang123
3 hours ago
0 replies
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circumcenter of BJK lies on line AC, median, right angle, circumcircle related
parmenides51   22
N 3 hours ago by Ilikeminecraft
Source: 2019 RMM Shortlist G1
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$.

Aleksandr Kuznetsov, Russia
22 replies
parmenides51
Jun 18, 2020
Ilikeminecraft
3 hours ago
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Squares on harmonic quadrilateral
Tkn   0
3 hours ago
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 $\overleftrightarrow{OX}$.
0 replies
Tkn
3 hours ago
0 replies
J
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Hard hyperbolic inverse
RenheMiResembleRice   2
N 3 hours ago by LawofCosine
Let f on the real numbers be defined by
$f\left(x\right)=x^{3}+\sinh x+1$
Explain why f has a differentiable inverse g.
2 replies
RenheMiResembleRice
5 hours ago
LawofCosine
3 hours ago
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\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{
sqing   2
N Jul 26, 2024 by sqing
Source: Own
Let $ a,b,c>0 $ and $ a\geq b+c.$ Prove that
$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{5}}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+2a^2}+\frac{c^2}{2a^2+b^2}} \geq\frac{\sqrt{2}}{3}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{2c^2+a^2}+\frac{c^2}{a^2+2b^2}} \geq\frac{1}{\sqrt{3}}$$
2 replies
sqing
Jul 25, 2024
sqing
Jul 26, 2024
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\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{
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Reveal topic tags
inequalities proposed
algebra
inequalities
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Source: Own
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sqing
41020 posts
sqing
#1 h  Jul 25, 2024, 4:31 AM • 1 Y
Y by GeoKing
Let $ a,b,c>0 $ and $ a\geq b+c.$ Prove that
$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+a^2}+\frac{c^2}{a^2+b^2}} \geq\sqrt{\frac{2}{5}}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{c^2+2a^2}+\frac{c^2}{2a^2+b^2}} \geq\frac{\sqrt{2}}{3}$$$$\frac{a}{b+c}\sqrt{ \frac{b^2}{2c^2+a^2}+\frac{c^2}{a^2+2b^2}} \geq\frac{1}{\sqrt{3}}$$
This post has been edited 1 time. Last edited by sqing, Jul 25, 2024, 8:57 AM
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frc
37 posts
frc
#2 h  Jul 26, 2024, 3:11 AM
Y by
https://artofproblemsolving.com/community/c6h3365546_fraca2_b2c2__leftfracb2c2a2fracc2a2b2_rightgeq
Notice$\frac{a^2}{(b+c)^2}\geq \frac{a^2}{2(b^2+c^2)}$
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sqing
41020 posts
sqing
#3 h  Jul 26, 2024, 3:38 AM
Y by
Very good.Thanks.
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