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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
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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
J
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Polynomial application with complex number
RenheMiResembleRice   1
N 6 minutes ago by Mathzeus1024
$P\left(x\right)=128x^{4}-32x^{2}+1$

By examining the roots of P(x), find the exact value of $\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)$
1 reply
RenheMiResembleRice
32 minutes ago
Mathzeus1024
6 minutes ago
J
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square geometry bisect $\angle ESB$
GorgonMathDota   12
N 8 minutes ago by AshAuktober
Source: BMO SL 2019, G1
Let $ABCD$ be a square of center $O$ and let $M$ be the symmetric of the point $B$ with respect to point $A$. Let $E$ be the intersection of $CM$ and $BD$, and let $S$ be the intersection of $MO$ and $AE$. Show that $SO$ is the angle bisector of $\angle ESB$.
12 replies
GorgonMathDota
Nov 8, 2020
AshAuktober
8 minutes ago
J
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Number of modular sequences with different residues
PerfectPlayer   1
N 24 minutes ago by Z4ADies
Source: Turkey TST 2025 Day 3 P9
Let \(n\) be a positive integer. For every positive integer $1 \leq k \leq n$ the sequence ${\displaystyle {\{ a_{i}+ki\}}_{i=1}^{n }}$ is defined, where $a_1,a_2, \dots ,a_n$ are integers. Among these \(n\) sequences, for at most how many of them does all the elements of the sequence give different remainders when divided by \(n\)?
1 reply
1 viewing
PerfectPlayer
6 hours ago
Z4ADies
24 minutes ago
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D1010 : How it is possible ?
Dattier   13
N an hour 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
13 replies
Dattier
Mar 10, 2025
Dattier
an hour ago
J
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Interesting inequality
sqing   1
N an hour ago by sqing
Source: Own
Let $ a,b\geq 2 . $ Prove that
$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq 27$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+\frac{81}{4}a^2b^2 \leq 189$$$$ (a^2-1)(b^2-1)(1-a^2b^2 )+ 162 ab \leq 513$$$$ (a^2-1)(b^2-1) (1-a^2b^2 )+21 a^2b^2\leq \frac{3219}{16}$$$$ (a^2-1)(b^2-1) (1-ab)+\frac{27}{8}a^2b^2\leq\frac{415+61\sqrt{61}}{18}$$
1 reply
sqing
2 hours ago
sqing
an hour ago
J
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Minimal Grouping in a Complete Graph
swynca   1
N an hour ago by swynca
Source: 2025 Turkey TST P1
In a complete graph with $2025$ vertices, each edge has one of the colors $r_1$, $r_2$, or $r_3$. For each $i = 1,2,3$, if the $2025$ vertices can be divided into $a_i$ groups such that any two vertices connected by an edge of color $r_i$ are in different groups, find the minimum possible value of $a_1 + a_2 + a_3$.
1 reply
1 viewing
swynca
4 hours ago
swynca
an hour ago
J
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Nice FE as the First Day Finale
swynca   1
N an hour ago by swynca
Source: 2025 Turkey TST P3
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x,y \in \mathbb{R}-\{0\}$,
$$ f(x) \neq 0 \text{ and } \frac{f(x)}{f(y)} + \frac{f(y)}{f(x)} - f \left( \frac{x}{y}-\frac{y}{x} \right) =2 $$
1 reply
swynca
4 hours ago
swynca
an hour ago
J
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Cn/lnn bound for S
EthanWYX2009   0
2 hours ago
Source: 2025 March 谜之竞赛-2
Prove that there exists an constant $C,$ such that for all integer $n\ge 2$ and a subset $S$ of $[n],$ satisfy $a\mid\tbinom ab$ for all $a,b\in S,$ $a>b,$ then $|S|\le \frac{Cn}{\ln n}.$

Created by Yuxing Ye
0 replies
2 viewing
EthanWYX2009
2 hours ago
0 replies
J
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Natural function and cubelike expression
sarjinius   2
N 2 hours ago by Kaimiaku
Source: Philippine Mathematical Olympiad 2025 P8
Let $\mathbb{N}$ be the set of positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for all $m, n \in \mathbb{N}$, \[m^2f(m) + n^2f(n) + 3mn(m + n)\]is a perfect cube.
2 replies
sarjinius
Mar 9, 2025
Kaimiaku
2 hours ago
J
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hard problem
Noname23   3
N 2 hours ago by Noname23
problem
3 replies
Noname23
Sunday at 4:57 PM
Noname23
2 hours ago
J
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Roots, bounding and other delusions
anantmudgal09   28
N 2 hours ago by kes0716
Source: INMO 2021 Problem 6
Let $\mathbb{R}[x]$ be the set of all polynomials with real coefficients. Find all functions $f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$ satisfying the following conditions:

[list]
[*] $f$ maps the zero polynomial to itself,
[*] for any non-zero polynomial $P \in \mathbb{R}[x]$, $\text{deg} \, f(P) \le 1+ \text{deg} \, P$, and
[*] for any two polynomials $P, Q \in \mathbb{R}[x]$, the polynomials $P-f(Q)$ and $Q-f(P)$ have the same set of real roots.
[/list]

Proposed by Anant Mudgal, Sutanay Bhattacharya, Pulkit Sinha
28 replies
anantmudgal09
Mar 7, 2021
kes0716
2 hours ago
J
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Inspired by my own results
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a , b $ be reals such that $ a+b+ab=1. $ Show that$$ 1-\frac{1 }{\sqrt2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2} $$Let $ a , b\geq 0 $ and $ a+b+ab=1. $ Show that$$ \frac{3}{2}\le \frac{1}{a^2+1}+\frac{1}{b^2+1}\le 1+\frac{1 }{\sqrt2} $$
5 replies
sqing
Yesterday at 8:32 AM
sqing
2 hours ago
J
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Polygon formed by the edges of an infinite chessboard
AlperenINAN   1
N 2 hours ago by AlperenINAN
Source: Turkey TST 2025 P5
Let $P$ be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers $a_1,a_2,a_3$ represent the number of unit squares that have exactly $1,2\text{ or } 3$ edges on the boundary of $P$ respectively. Find the largest real number $k$ such that the inequality $a_1+a_2>ka_3$ holds for each polygon constructed with these conditions.
1 reply
AlperenINAN
4 hours ago
AlperenINAN
2 hours ago
J
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Interesting inequality
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -6ab\geq-15$$$$(a^2-1)(b^2-1) -7ab\geq -\frac{58}{3}$$$$(a^3-1)(b^3-1) -\frac{21}{4}a^2b^2\geq -35$$$$(a^3-1)(b^3-1) -6a^2b^2\geq-\frac{2391}{49}$$
5 replies
sqing
6 hours ago
sqing
2 hours ago
J
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J
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Interesting inequality
sqing   5
N 2 hours ago by sqing
Source: Own
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -6ab\geq-15$$$$(a^2-1)(b^2-1) -7ab\geq -\frac{58}{3}$$$$(a^3-1)(b^3-1) -\frac{21}{4}a^2b^2\geq -35$$$$(a^3-1)(b^3-1) -6a^2b^2\geq-\frac{2391}{49}$$
5 replies
sqing
6 hours ago
sqing
2 hours ago
J
Interesting inequality
G H J
Reveal topic tags
inequalities proposed
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Own
The post below has been deleted. Click to close.
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sqing
41063 posts
sqing
#1 h  6 hours ago
Y by
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -6ab\geq-15$$$$(a^2-1)(b^2-1) -7ab\geq -\frac{58}{3}$$$$(a^3-1)(b^3-1) -\frac{21}{4}a^2b^2\geq -35$$$$(a^3-1)(b^3-1) -6a^2b^2\geq-\frac{2391}{49}$$
This post has been edited 2 times. Last edited by sqing, 6 hours ago
Z K Y
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sqing
41063 posts
sqing
#2 h  5 hours ago
Y by
Let $ a,b\geq 2 . $ Prove that
$$(1-a^2)(1-b^2) (1-ab)+27ab\leq 81$$$$(1-a^2)(1-b^2) (1-ab)+28ab\leq \frac{133+625\sqrt 5}{18}$$$$(1-a^3)(1-b^3) (1-ab)+175ab\leq 553$$$$ (1-a^3)(1-b^3) (1-ab)+21a^2b^2 \leq 189$$$$ (1-a^3)(1-b^3) (1-a^2b^2)+1022ab \leq 3353$$$$ (1-a^3)(1-b^3) (1-a^2b^2)+ \frac{511}{4} a^2b^2 \leq 1309$$
This post has been edited 2 times. Last edited by sqing, 2 hours ago
Z K Y
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lbh_qys
399 posts
lbh_qys
#3 h  5 hours ago
Y by
sqing wrote:
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -6ab\geq-15$$

$$(a^2 - 1)(b^2 - 1) - 6ab \geq (4a-5)(4b-5) - 6ab = 10(a-2)(b-2) - 15 \geq -15$$
Z K Y
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sqing
41063 posts
sqing
#4 h  5 hours ago
Y by
Very nice.Thank lbh_qys.
Z K Y
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lbh_qys
399 posts
lbh_qys
#5 h  5 hours ago
Y by
sqing wrote:
Let $ a,b\geq 2 . $ Prove that
$$(a^2-1)(b^2-1) -7ab\geq -\frac{58}{3}$$

$$\mathrm{LHS} - \mathrm{RHS} = \frac{(3a+3b-13)^2}{3} + ( ab + 2a + 2b - 9)(a-2)(b-2) \geq 0$$
Z K Y
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sqing
41063 posts
sqing
#6 h  2 hours ago
Y by
Very nice.Thank lbh_qys.
Z K Y
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