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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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Function equation
Dynic   1
N 2 minutes ago by jasperE3
Find all function $f:\mathbb{Z}\to\mathbb{Z}$ satisfy all conditions below:
i) $f(n+1)>f(n)$ for all $n\in \mathbb{Z}$
ii) $f(-n)=-f(n)$ for all $n\in \mathbb{Z}$
iii) $f(a^3+b^3+c^3+d^3)=f^3(a)+f^3(b)+f^3(c)+f^3(d)$ for all $n\in \mathbb{Z}$
1 reply
Dynic
an hour ago
jasperE3
2 minutes ago
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Geometry
srnjbr   0
7 minutes ago
in triangle abc, we know that bac=60. the circumcircle of the center i is tangent to the sides ab and ac at points e and f respectively. the midpoint of side bc is called m. if lines bi and ci intersect line ef at points p and q respectively, show that pmq is equilateral.
0 replies
srnjbr
7 minutes ago
0 replies
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JBMO Shortlist 2021 N1
Lukaluce   14
N 8 minutes ago by ali123456
Source: JBMO Shortlist 2021
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to
factorials of some positive integers.

Proposed by Nikola Velov, Macedonia
14 replies
Lukaluce
Jul 2, 2022
ali123456
8 minutes ago
J
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Very easy inequality
pggp   2
N 12 minutes ago by ali123456
Source: Polish Junior MO Second Round 2019
Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.
2 replies
pggp
Oct 26, 2020
ali123456
12 minutes ago
J
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Problem about Euler's function
luutrongphuc   1
N 13 minutes ago by ishan.panpaliya
Prove that for every integer $n \ge 5$, we have:
$$ 2^{n^2+3n-13} \mid \phi \left(2^{2^{n}}-1 \right)$$
1 reply
luutrongphuc
2 hours ago
ishan.panpaliya
13 minutes ago
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Problem 5
blug   1
N 34 minutes ago by WallyWalrus
Source: Polish Junior Math Olympiad Finals 2025
Each square on a 5×5 board contains an arrow pointing up, down, left, or right. Show that it is possible to remove exactly 20 arrows from this board so that no two of the remaining five arrows point to the same square.
1 reply
blug
Mar 15, 2025
WallyWalrus
34 minutes ago
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Cool Number Theory
Fermat_Fanatic108   6
N an hour ago by epl1
For an integer with 5 digits $n=abcde$ (where $a, b, c, d, e$ are the digits and $a\neq 0$) we define the \textit{permutation sum} as the value $$bcdea+cdeab+deabc+eabcd$$For example the permutation sum of 20253 is $$02532+25320+53202+32025=113079$$Let $m$ and $n$ be two fivedigit integers with the same permutation sum.
Prove that $m=n$.
6 replies
Fermat_Fanatic108
5 hours ago
epl1
an hour ago
J
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Incenter geometry with parallel lines
nAalniaOMliO   1
N an hour ago by LenaEnjoyer
Source: Belarusian MO 2023
Let $\omega$ be the incircle of triangle $ABC$. Line $l_b$ is parallel to side $AC$ and tangent to $\omega$. Line $l_c$ is parallel to side $BC$ and tangent to $\omega$. It turned out that the intersection point of $l_b$ and $l_c$ lies on circumcircle of $ABC$
Find all possible values of $\frac{AB+AC}{BC}$
1 reply
nAalniaOMliO
Apr 16, 2024
LenaEnjoyer
an hour ago
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Problem for VASC, SI Book
hungkhtn   21
N 2 hours ago by imnotgoodatmathsorry
Source: please let him prove it first
Let $a,b,c$ be non-negative real numbers such that $a+b+c=3$. Prove that
\[a\sqrt{1+b^{3}}+b\sqrt{1+c^{3}}+c\sqrt{1+a^{3}}\le 5.\]
21 replies
hungkhtn
Jun 5, 2007
imnotgoodatmathsorry
2 hours ago
J
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IMO Shortlist 2009 - Problem N4
April   12
N 2 hours ago by asdf334
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$.

Proposed by North Korea
12 replies
April
Jul 5, 2010
asdf334
2 hours ago
J
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China Team Selection Test 2015 TST 1 Day 2 Q1
sqing   6
N 2 hours ago by sttsmet
Source: China Hangzhou
Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.
6 replies
sqing
Mar 14, 2015
sttsmet
2 hours ago
J
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China Mathematical Olympiad 1993 problem5
jred   3
N 2 hours ago by iStud
Source: China Mathematical Olympiad 1993 problem5
$10$ students bought some books in a bookstore. It is known that every student bought exactly three kinds of books, and any two of them shared at least one kind of book. Determine, with proof, how many students bought the most popular book at least? (Note: the most popular book means most students bought this kind of book)
3 replies
jred
Sep 23, 2013
iStud
2 hours ago
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x and o game, in an infinite grid of regular triangles
parmenides51   5
N 3 hours ago by Lil_flip38
Source: Norwegian Mathematical Olympiad 2017 - Abel Competition p3b
In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up.
Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game.
Determine whether one of the players can force a victory.
IMAGE
5 replies
parmenides51
Sep 3, 2019
Lil_flip38
3 hours ago
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BMN is equilateral iff rectangle ABCD is square
parmenides51   4
N 3 hours ago by Tsikaloudakis
Source: 2004 Romania NMO SL - Shortlist VII-VIII p8 https://artofproblemsolving.com/community/c3950157_
Consider a point $M$ on the diagonal $BD$ of a given rectangle $ABCD$, such that $\angle AMC = \angle CMD$. The point $N$ is the intersection point between $AM$ and the parallel line to $CM$ that contains $B$. Prove that the triangle $BMN$ is equilateral if and only if $ABCD$ is a square.

Valentin Vornicu
4 replies
parmenides51
Sep 16, 2024
Tsikaloudakis
3 hours ago
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J
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Strike the inequality
giangtruong13   2
N Mar 16, 2025 by sqing
Source: Idk
Let $a,b,c \geq 0$ satisfy that $a+b+c=3$. Prove that $$\sum a\sqrt{b^3+1} \leq 5$$
2 replies
giangtruong13
Mar 15, 2025
sqing
Mar 16, 2025
J
Strike the inequality
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Reveal topic tags
inequalities
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Source: Idk
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giangtruong13
68 posts
giangtruong13
#1 h  Mar 15, 2025, 4:24 PM
Y by
Let $a,b,c \geq 0$ satisfy that $a+b+c=3$. Prove that $$\sum a\sqrt{b^3+1} \leq 5$$
This post has been edited 1 time. Last edited by giangtruong13, Mar 15, 2025, 4:25 PM
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arqady
30150 posts
arqady
#2 h  Mar 15, 2025, 8:42 PM
Y by
giangtruong13 wrote:
Let $a,b,c \geq 0$ satisfy that $a+b+c=3$. Prove that $$\sum a\sqrt{b^3+1} \leq 5$$
Use AM-GM and Rearrangement.
Z K Y
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sqing
41104 posts
sqing
#3 h  Mar 16, 2025, 1:11 AM
Y by
giangtruong13 wrote:
Let $a,b,c \geq 0$ satisfy that $a+b+c=3$. Prove that $$\sum a\sqrt{b^3+1} \leq 5$$
https://artofproblemsolving.com/community/c6h152446p857973
https://artofproblemsolving.com/community/c6h514418p2890042
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