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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:
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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
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Classic algebra problem
Makaveli   12
N 10 minutes ago by RedFireTruck
Source: USAMO1994
Let $\, k_1 < k_2 < k_3 < \cdots \,$ be positive integers, no two consecutive, and let $\, s_m = k_1 + k_2 + \cdots + k_m \,$ for $\, m = 1,2,3, \ldots \; \;$. Prove that, for each positive integer $\, n, \,$ the interval $\, [s_n, s_{n+1}) \,$ contains at least one perfect square.
12 replies
+1 w
Makaveli
Aug 22, 2005
RedFireTruck
10 minutes ago
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The return of a legend inequality
giangtruong13   5
N 22 minutes ago by ehuseyinyigit
Source: Legacy
Given that $0<a,b,c,d<1$.Prove that: $$ (1-a)(1-b)(1-c)(1-d) > 1-a-b-c-d $$
5 replies
giangtruong13
5 hours ago
ehuseyinyigit
22 minutes ago
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concurrency wanted, starting with a 45-135/2 - 135/2 triangle
parmenides51   2
N an hour ago by GioOrnikapa
Source: 2020 Balkan MO shortlist G5
Let $ABC$ be an isosceles triangle with $AB = AC$ and $\angle A = 45^o$. Its circumcircle $(c)$ has center $O, M$ is the midpoint of $BC$ and $D$ is the foot of the perpendicular from $C$ to $AB$. With center $C$ and radius $CD$ we draw a circle which internally intersects $AC$ at the point $F$ and the circle $(c)$ at the points $Z$ and $E$, such that $Z$ lies on the small arc $BC$ and $E$ on the small arc $AC$. Prove that the lines $ZE$, $CO$, $FM$ are concurrent.

Brazitikos Silouanos, Greece
2 replies
parmenides51
Sep 14, 2021
GioOrnikapa
an hour ago
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Another SL problem about fibonacci numbers :3
MathLuis   13
N an hour ago by hgomamogh
Source: ISL 2020 C4
The Fibonacci numbers $F_0, F_1, F_2, . . .$ are defined inductively by $F_0=0, F_1=1$, and $F_{n+1}=F_n+F_{n-1}$ for $n \ge 1$. Given an integer $n \ge 2$, determine the smallest size of a set $S$ of integers such that for every $k=2, 3, . . . , n$ there exist some $x, y \in S$ such that $x-y=F_k$.

Proposed by Croatia
13 replies
MathLuis
Jul 20, 2021
hgomamogh
an hour ago
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USAMO 1983 Problem 2 - Roots of Quintic
Binomial-theorem   28
N 2 hours ago by Marcus_Zhang
Source: USAMO 1983 Problem 2
Prove that the roots of\[x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\] cannot all be real if $2a^2 < 5b$.
28 replies
1 viewing
Binomial-theorem
Aug 16, 2011
Marcus_Zhang
2 hours ago
J
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Find the period
Anto0110   0
2 hours ago
Let $a_1, a_2, ..., a_k, ...$ be a sequence that consists of an initial block of $p$ positive distinct integers that then repeat periodically. This means that $\{a_1, a_2, \dots, a_p\}$ are $p$ distinct positive integers and $a_{n+p}=a_n$ for every positive integer $n$. The terms of the sequence are not known and the goal is to find the period $p$. To do this, at each move it possible to reveal the value of a term of the sequence at your choice.
If $p$ is one of the first $k$ prime numbers, find for which values of $k$ there exist a strategy that allows to find $p$ revealing at most $8$ terms of the sequence.
0 replies
Anto0110
2 hours ago
0 replies
J
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circle geometry showing perpendicularity
Kyj9981   2
N 2 hours ago by Double07
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. A line through $B$ intersects $\omega_1$ and $\omega_2$ at points $C$ and $D$, respectively. Line $AD$ intersects $\omega_1$ at point $E \neq A$, and line $AC$ intersects $\omega_2$ at point $F \neq A$. If $O$ is the circumcenter of $\triangle AEF$, prove that $OB \perp CD$.
2 replies
Kyj9981
Today at 11:53 AM
Double07
2 hours ago
J
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Diophantine equation
PaperMath   9
N 2 hours ago by gaussiemann144
Find the $5$ smallest positive solutions of $x$ that has an integer $k$ that satisfies $x^2=3k^2+4$
9 replies
PaperMath
Mar 12, 2025
gaussiemann144
2 hours ago
J
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How many ordered pairs of numbers can we find?
BR1F1SZ   2
N 2 hours ago by BR1F1SZ
Source: 2024 Argentina TST P6
Given $2024$ non-negative real numbers $x_1, x_2, \dots, x_{2024}$ that satisfy $x_1 + x_2 + \cdots + x_{2024} = 1$, determine the maximum possible number of ordered pairs $(i, j)$ such that
\[ x_i^2 + x_j \geqslant \frac{1}{2023}. \]
2 replies
BR1F1SZ
Jan 25, 2025
BR1F1SZ
2 hours ago
J
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IMO 2009, Problem 5
orl   87
N 2 hours ago by asdf334
Source: IMO 2009, Problem 5
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b + f(a) - 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)

Proposed by Bruno Le Floch, France
87 replies
orl
Jul 16, 2009
asdf334
2 hours ago
J
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1978 USAMO #1
Mrdavid445   54
N 3 hours ago by Marcus_Zhang
Given that $a,b,c,d,e$ are real numbers such that

$a+b+c+d+e=8$,
$a^2+b^2+c^2+d^2+e^2=16$.

Determine the maximum value of $e$.
54 replies
Mrdavid445
Aug 16, 2011
Marcus_Zhang
3 hours ago
J
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degree of f=2^k
Sayan   15
N 3 hours ago by Gejabsk
Source: ISI 2012 #8
Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$.

i) Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$.

ii) Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.
15 replies
Sayan
May 13, 2012
Gejabsk
3 hours ago
J
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Local-global with Fibonacci numbers
MarkBcc168   26
N 3 hours ago by pi271828
Source: ELMO 2020 P2
Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number?

Proposed by Fedir Yudin.
26 replies
MarkBcc168
Jul 28, 2020
pi271828
3 hours ago
J
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Cauchy functional equations
syk0526   10
N 3 hours ago by imagien_bad
Source: FKMO 2013 #2
Find all functions $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions.
(a) $ f(x) \ge 0 $ for all $ x \in \mathbb{R} $.
(b) For $ a, b, c, d \in \mathbb{R} $ with $ ab + bc + cd = 0 $, equality $ f(a-b) + f(c-d) = f(a) + f(b+c) + f(d) $ holds.
10 replies
syk0526
Mar 24, 2013
imagien_bad
3 hours ago
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J
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A two-variable & non-homogenous inequality that seems hard to me
MyLifeMyChoice   3
N Mar 15, 2025 by Radin_
Source: Developing from a larger, three-variable one
For $a,b>0$, prove/disprove the following claim: :maybe:

$a^3b^3+\frac{1}{a^3}+\frac{1}{b^3}+3\stackrel{?}{\ge}a^2b+b^2a+\frac{1}{a^2b}+\frac{1}{b^2a}+\frac{a}{b}+\frac{b}{a}$
3 replies
MyLifeMyChoice
Mar 13, 2025
Radin_
Mar 15, 2025
J
A two-variable & non-homogenous inequality that seems hard to me
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Reveal topic tags
inequalities
hint
advice
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G H BBookmark kLocked kLocked NReply
Source: Developing from a larger, three-variable one
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MyLifeMyChoice
50 posts
MyLifeMyChoice
#1 h  Mar 13, 2025, 5:27 AM
Y by
For $a,b>0$, prove/disprove the following claim: :maybe:

$a^3b^3+\frac{1}{a^3}+\frac{1}{b^3}+3\stackrel{?}{\ge}a^2b+b^2a+\frac{1}{a^2b}+\frac{1}{b^2a}+\frac{a}{b}+\frac{b}{a}$
This post has been edited 1 time. Last edited by MyLifeMyChoice, Mar 15, 2025, 6:54 PM
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DKI
27 posts
DKI
#2 h  Mar 13, 2025, 8:57 AM • 1 Y
Y by MyLifeMyChoice
Equivalent to Schur's inequality.
Z K Y
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MyLifeMyChoice
50 posts
MyLifeMyChoice
#3 h  Mar 15, 2025, 6:56 PM
Y by
DKI wrote:
Equivalent to Schur's inequality.

I haven't get it yet dear friend. Could you please explain your observation just a little bit :huh:
This post has been edited 2 times. Last edited by MyLifeMyChoice, Mar 15, 2025, 7:00 PM
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Radin_
1 post
Radin_
#4 h  Mar 15, 2025, 7:45 PM • 1 Y
Y by ehuseyinyigit
by Schur's inequality we have that: \[a^3+b^3+c^3+3abc \ge a^2 b+a^2 c+b^2 a+b^2 c+c^2 a+c^2 b\]now simply replace a with $\frac{1}{a}$, b with $\frac{1}{b}$ and c with ab to get the desired result
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