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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.
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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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Flo0r functi0n
m4thbl3nd3r   5
N 15 minutes ago by pco
Find all positive integers such that $$n=\lfloor \sqrt{n}\rfloor^2+\lfloor \sqrt{n}\rfloor$$
5 replies
m4thbl3nd3r
2 hours ago
pco
15 minutes ago
J
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Hard T^T
Noname23   1
N 17 minutes ago by RagvaloD
<problem>
1 reply
Noname23
44 minutes ago
RagvaloD
17 minutes ago
J
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At least 3 co-prime pairs
eezad3   0
18 minutes ago
You are given a random permutation of the first $n$ integers where $n>10$. Show that there exists at least three positions $i$ such that $gcd(p[i], p[i+1])=1$ (here $p[i]$ refers to the $i$-th position integer in the permutation)
0 replies
eezad3
18 minutes ago
0 replies
J
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Suspicious Quadrilateral Geometry
YaoAOPS   4
N 18 minutes ago by Giabach298
Source: 2025 CTST P8
Let quadrilateral $A_1A_2A_3A_4$ be not cyclic and haves edges not parallel to each other.

Denote $B_i$ as the intersection of the tangent line at $A_i$ with respect to circle $A_{i-1}A_iA_{i+1}$ and the $A_{i+2}$-symmedian with respect to triangle $A_{i+1}A_{i+2}A_{i+3}$ and $C_i$ as the intersection of lines $A_iA_{i+1}$ and $B_iB_{i+1}$, where all indexes taken cyclically.

Prove that $C_1$, $C_2$, $C_3$, and $C_4$ are collinear.
4 replies
YaoAOPS
Mar 10, 2025
Giabach298
18 minutes ago
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i need help
MR.1   0
an hour ago
Source: help
can you guys tell me problems about fe in $R+$(i know $R$ well). i want to study so if you guys have some easy or normal problems please send me
0 replies
MR.1
an hour ago
0 replies
J
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Changeable polynomials, can they ever become equal?
mshtand1   4
N an hour ago by CHESSR1DER
Source: Ukrainian Mathematical Olympiad 2025. Day 2, Problem 11.5
Initially, two constant polynomials are written on the board: \(0\) and \(1\). At each step, it is allowed to add \(1\) to one of the polynomials and to multiply another one by the polynomial \(45x + 2025\). Can the polynomials become equal at some point?

Proposed by Oleksii Masalitin
4 replies
mshtand1
Yesterday at 12:47 AM
CHESSR1DER
an hour ago
J
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Guessing polynomial by its maximum values on segments
NO_SQUARES   3
N an hour ago by pco
Source: Kvant 2025 no. 1 M2828 and The XIX Southern Mathematical Tournament
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
M. Didin
3 replies
NO_SQUARES
Yesterday at 3:21 PM
pco
an hour ago
J
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easy number theory
MuradSafarli   1
N an hour ago by Tuvshuu
\[ v_p(n!) \leq \frac{n}{p - 1} \]
1 reply
MuradSafarli
2 hours ago
Tuvshuu
an hour ago
J
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Inspired by Kazakhstan 2017
sqing   1
N 2 hours ago by sqing
Source: Own
Let $a,b,c\ge \frac{1}{2}$ and $a+b+c=2. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 1$$Let $a,b,c\ge \frac{1}{3}$ and $a+b+c=1. $ Prove that
$$\left(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}\right)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right)\ge 9$$
1 reply
sqing
3 hours ago
sqing
2 hours ago
J
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Collinear geometry problem with incircle
ilovemath0402   1
N 2 hours ago by deraxenrovalo
Given acute $\triangle ABC$ not isosceles, the incircle $(I)$. $D,E,F$ is the intersection of $(I)$ with $BC,CA,AB$. $P$ is the projection of $D$ onto $EF$. $DP$ cut $(I)$ at the second point $K$. $L$ is the projection of $A$ onto $IK$. $(LEC), (LFB)$ cut $(I)$ at the second point $M,N$ respectively. Prove $M,N,P$ are collinear
1 reply
ilovemath0402
Jul 22, 2023
deraxenrovalo
2 hours ago
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Wait wasn&#039;t it the reciprocal in the paper?
Supercali   6
N 3 hours ago by kes0716
Source: India TST 2023 Day 2 P1
Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$.

Proposed by Navilarekallu Tejaswi
6 replies
Supercali
Jul 9, 2023
kes0716
3 hours ago
J
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About old Inequality
perfect_square   0
3 hours ago
Source: Arqady
This is: $a,b,c>0$ which satisfy $abc=1$
Prove that: $ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$
By $ uvw $ method, I can assum $b=c=x,a=\frac{1}{x^2}$
But I can't prove:
$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $
Is there an another way?
0 replies
perfect_square
3 hours ago
0 replies
J
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inquality
karasuno   1
N 4 hours ago by sqing
The real numbers $x,y,z \ge \frac{1}{2}$ are given such that $x^{2}+y^{2}+z^{2}=1$. Prove the inequality $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2 .$$
1 reply
karasuno
4 hours ago
sqing
4 hours ago
J
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Number Theory
karasuno   0
4 hours ago
Solve the equation $$n!+10^{2014}=m^{4}$$in natural numbers m and n.
0 replies
karasuno
4 hours ago
0 replies
J
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J
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Hardest problem on Kyiv City MO 2021. Possibly ever
MS_Kekas   1
N Dec 22, 2023 by anhduy98
Source: Kyiv City MO 2021 Round 1, Problem 10.1
Prove the following inequality:

$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
Proposed by Oleksii Masalitin
1 reply
MS_Kekas
Dec 21, 2023
anhduy98
Dec 22, 2023
J
Hardest problem on Kyiv City MO 2021. Possibly ever
G H J
Reveal topic tags
inequalities
L
G H BBookmark kLocked kLocked NReply
Source: Kyiv City MO 2021 Round 1, Problem 10.1
The post below has been deleted. Click to close.
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MS_Kekas
275 posts
MS_Kekas
#1 h  Dec 21, 2023, 6:54 PM
Y by
Prove the following inequality:

$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
Proposed by Oleksii Masalitin
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
anhduy98
1080 posts
anhduy98
#2 h  Dec 22, 2023, 12:36 AM
Y by
MS_Kekas wrote:
Prove the following inequality:

$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
Proposed by Oleksii Masalitin

$$LHS=\frac{2\sin{1}.\sin{1} + 2\sin{3}.\sin{1} + \ldots +2 \sin{2021}.\sin{1}}{2\sin{1}}$$$$LHS=\frac{1-\cos{2} +\cos{2}-\cos{4} + \ldots +\cos{2020}-\cos{2022}}{2\sin{1}}=\frac{1-\cos{2022}}{2\sin{1}}=\frac{\sin^2{1011}}{\sin{1}}$$$$LHS=\frac{\sin^2{1011}}{\sin{1}} >\frac{\sin^2{1011}}{\sin{\dfrac{\pi}{3}}}=RHS.$$
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