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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!
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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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Number theory question with many (confusing) variables
urfinalopp   1
N 9 minutes ago by mathprodigy2011
Given m,n,p,q \in \mathbb{N+}, find all solutions to 2^{m}3^{n}+5^{p}=7^{q}$

One of the paths I've found is to boil it down to solving two non-simultaneous equations 2^{m_1}+5^{n_1}=7^{q_1} and
7^{m_1}+5^{n_1}=2^{q_1} but its too hard. Any other approaches/solutions or a continuation of this path?
1 reply
urfinalopp
2 hours ago
mathprodigy2011
9 minutes ago
J
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funny title
nguyenvana   1
N 24 minutes ago by pco
Source: no from book
Find all the functions f: R+ to R+ which satisfy the functional equation:
f(2f(x)+f(y)+xy)=xy+2x+y (x,y R+)
1 reply
nguyenvana
2 hours ago
pco
24 minutes ago
J
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FB = BK , circumcircle and altitude related (In the World of Mathematics 516)
parmenides51   4
N 25 minutes ago by Nioronean
Let $BT$ be the altitude and $H$ be the intersection point of the altitudes of triangle $ABC$. Point $N$ is symmetric to $H$ with respect to $BC$. The circumcircle of triangle $ATN$ intersects $BC$ at points $F$ and $K$. Prove that $FB = BK$.

(V. Starodub, Kyiv)
4 replies
parmenides51
Apr 19, 2020
Nioronean
25 minutes ago
J
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subsets of subset has same sum
61plus   3
N 27 minutes ago by sttsmet
Source: 2015 China TST 2 Day 2 Q2
Set $S$ to be a subset of size $68$ of $\{1,2,...,2015\}$. Prove that there exist $3$ pairwise disjoint, non-empty subsets $A,B,C$ such that $|A|=|B|=|C|$ and $\sum_{a\in A}a=\sum_{b\in B}b=\sum_{c\in C}c$
3 replies
61plus
Mar 19, 2015
sttsmet
27 minutes ago
J
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Dear Sqing: So Many Inequalities...
hashtagmath   23
N 31 minutes ago by MTA_2024
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
23 replies
hashtagmath
Oct 30, 2024
MTA_2024
31 minutes ago
J
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(ab)^2 + (bc)^2 + (ca)^2
GorgonMathDota   13
N 32 minutes ago by ektorasmiliotis
Source: Shortlist BMO 2019, A5
Let $a,b,c$ be positive real numbers, such that $(ab)^2 + (bc)^2 + (ca)^2 = 3$. Prove that
\[ (a^2 - a + 1)(b^2 - b + 1)(c^2 - c + 1) \ge 1. \]
Proposed by Florin Stanescu (wer), România
13 replies
GorgonMathDota
Nov 7, 2020
ektorasmiliotis
32 minutes ago
J
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weird combinatorics/algebra
Dr.Poe98   1
N 37 minutes ago by americancheeseburger4281
Source: Brazil Cono Sur TST 2024 - T3/P2
For each natural number $n\ge3$, let $m(n)$ be the maximum number of points inside or on the sides of a regular $n$-agon of side $1$ such that the distance between any two points is greater than $1$. Prove that $m(n)\ge n$ for $n>6$.
1 reply
Dr.Poe98
Oct 21, 2024
americancheeseburger4281
37 minutes ago
J
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A nice problem
hanzo.ei   0
an hour ago

Given a nonzero real number \(a\) and a polynomial \(P(x)\) with real coefficients of degree \(n\) (\(n > 1\)) such that \(P(x)\) has no real roots. Prove that the polynomial
\[ Q(x) \;=\; P(x) \;+\; a\,P'(x) \;+\; a^2\,P''(x) \;+\; \dots \;+\; a^n\,P^{(n)}(x) \]has no real roots.
0 replies
hanzo.ei
an hour ago
0 replies
J
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interesting algebra/geometry
Dr.Poe98   2
N an hour ago by americancheeseburger4281
Source: Brazil Cono Sur TST 2024 - T2/P4
In the cartesian plane, consider the subset of all the points with both integer coordinates. Prove that it is possible to choose a finite non-empty subset $S$ of these points in such a way that any line $l$ that forms an angle of $90^{\circ},0^{\circ},135^{\circ}$ or $45^{\circ}$ with the positive horizontal semi-axis intersects $S$ at exactly $2024$ points or at no points.
2 replies
Dr.Poe98
Oct 21, 2024
americancheeseburger4281
an hour ago
J
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Round up to the nearest power of two
navi_09220114   1
N an hour ago by ja.
Source: Own. Malaysian IMO TST 2025 P6
A sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ is called good, if $a_i$ are non-negative integers, and $a_{i+1}-a_{i}$ is either $0$ or $1$ for all $1\le i\le m-1$.

Fix a positive integer $n$, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence $2^{a_1}, 2^{a_2}, \cdots,2^{a_m}$ that appears on the whiteboard, and then he writes the number $2^k$ such that $$2^{k-1}<2^{a_1}+2^{a_2}+\cdots+2^{a_m}\le 2^{k}$$Suppose Ivan starts with the least possible number of ones to obtain $2^n$ after some steps, determine the minimum number of steps he will need in order to do so.

Proposed by Ivan Chan Kai Chin
1 reply
navi_09220114
5 hours ago
ja.
an hour ago
J
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9 Three concurrent chords
v_Enhance   5
N an hour ago by MS_asdfgzxcvb
Three distinct circles $\Omega_1$, $\Omega_2$, $\Omega_3$ cut three common chords concurrent at $X$. Consider two distinct circles $\Gamma_1$, $\Gamma_2$ which are internally tangent to all $\Omega_i$. Determine, with proof, which of the following two statements is true.

(1) $X$ is the insimilicenter of $\Gamma_1$ and $\Gamma_2$
(2) $X$ is the exsimilicenter of $\Gamma_1$ and $\Gamma_2$.
5 replies
v_Enhance
Yesterday at 8:45 PM
MS_asdfgzxcvb
an hour ago
J
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A number theory about divisors which no one fully solved at the contest
nAalniaOMliO   4
N an hour ago by AL1296
Source: Belarusian national olympiad 2024
Let's call a pair of positive integers $(k,n)$ interesting if $n$ is composite and for every divisor $d<n$ of $n$ at least one of $d-k$ and $d+k$ is also a divisor of $n$
Find the number of interesting pairs $(k,n)$ with $k \leq 100$
M. Karpuk
4 replies
nAalniaOMliO
Jul 24, 2024
AL1296
an hour ago
J
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Inequalities
sqing   3
N 2 hours ago by sqing
Let $ a,b> 0$ and $ a+b=1 . $ Prove that
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{4+\frac{k}{4096}}{1+ ka^7b^7}$$Where $\frac{8192}{3}\geq k>0 .$
$$ \frac{1}{a}+\frac{1}{b}\geq \frac{\frac{14}{3}}{1+ \frac{8192}{3}a^7b^7}$$
3 replies
sqing
3 hours ago
sqing
2 hours ago
J
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Inequalities
sqing   0
2 hours ago
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that$$a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$$Equality holds when $ a=b=c=\frac{1}{3} $ or $ a=0,b=\frac{3}{4},c=\frac{1}{4} $ or $ a=\frac{1}{4} ,b=0,c=\frac{3}{4} $
or $ a=\frac{3}{4} ,b=\frac{1}{4},c=0. $
0 replies
sqing
2 hours ago
0 replies
J
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J
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Inequalities
sqing   4
N Yesterday at 2:57 AM by sqing
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
4 replies
sqing
Mar 19, 2025
sqing
Yesterday at 2:57 AM
J
Inequalities
G H J
Reveal topic tags
inequalities
L
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sqing
41172 posts
sqing
#1 h  Mar 19, 2025, 1:42 PM
Y by
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$$$(a^2-2)(b-1)(c^2-2) - \frac{3}{2}abc\geq -6$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
no_room_for_error
319 posts
no_room_for_error
#2 h  Mar 19, 2025, 6:07 PM
Y by
sqing wrote:
Let $ a,b,c\geq 2 . $ Prove that
$$(a-1)(b^2-2)(c-1) - \frac{1}{2}abc\geq -2$$

$$ \begin{aligned} (a-1)(b^2-2)(c-1) &\geq 2(a-1)(b-1)(c-1)\\ &=\frac{1}{2}abc-2+\frac{3}{2}(a-2)(b-2)(c-2)+(a-2)(b-2)+(b-2)(c-2)+(c-2)(a-2)\\ &\geq \frac{1}{2}abc-2\\ \end{aligned} $$
Equality occurs when $b=2$ and $(c-2)(a-2)=0$.
Z K Y
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sqing
41172 posts
sqing
#3 h  Mar 20, 2025, 2:19 AM
Y by
Very very nice.Thank no_room_for_error.
Z K Y
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sqing
41172 posts
sqing
#4 h  Mar 20, 2025, 1:58 PM
Y by
Let $ a,b\geq 0 $ and $ a^2+ab+b^2+a+b=5. $ Prove that
$$ \frac{ (a+b)(ab+1)}{a+b-1} \leq 4$$$$ \frac{ (a+b)(ab+a+b)}{a+b-1} \leq 6$$$$ \frac{a^2b+b^2+a+1}{a+b } \leq \frac{3\sqrt{21}-2}{5}$$$$ \frac{a^2b+b^2+a }{a+b+1} \leq \frac{3\sqrt{21}-8}{5}$$
Z K Y
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sqing
41172 posts
sqing
#5 h  Yesterday at 2:57 AM
Y by
Let $ a,b,c\geq \frac{4}{5}$ and $ a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+8 . $ Prove that
$$ ab+bc +ca\leq 27 $$
Z K Y
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