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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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0 replies
jlacosta
Apr 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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Sum and product of 5 numbers
jl_   1
N 2 minutes ago by jl_
Source: Malaysia IMONST 2 2023 (Primary) P2
Ivan bought $50$ cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number $100000$. How many cats of each breed does he have?
1 reply
jl_
3 hours ago
jl_
2 minutes ago
J
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Interesting inequalities
sqing   3
N 5 minutes ago by sqing
Source: Own
Let $ a,b> 0 $ and $ a+b\leq 2ab . $ Prove that
$$\frac{ 9a^2- ab +9b^2 }{ a^2(1+b^4)}\leq\frac{17 }{2}$$$$\frac{a- ab+b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$$$\frac{2a- 3ab+2b }{ a^2(1+b^4)}\leq\frac{1 }{2}$$
3 replies
sqing
4 hours ago
sqing
5 minutes ago
J
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a+b+c=2 ine
KhuongTrang   30
N 9 minutes ago by KhuongTrang
Source: own
Problem. Given non-negative real numbers $a,b,c: ab+bc+ca>0$ satisfying $a+b+c=2.$ Prove that $$\color{blue}{\frac{a}{\sqrt{2a+3bc}}+\frac{b}{\sqrt{2b+3ca}}+\frac{c}{\sqrt{2c+3ab}} \le \sqrt{\frac{2}{ab+bc+ca}}. }$$
30 replies
KhuongTrang
Jun 25, 2024
KhuongTrang
9 minutes ago
J
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Divisibility holds for all naturals
XbenX   12
N 36 minutes ago by Null314
Source: 2018 Balkan MO Shortlist N5
Let $x,y$ be positive integers. If for each positive integer $n$ we have that $$(ny)^2+1\mid x^{\varphi(n)}-1.$$Prove that $x=1$.

(Silouanos Brazitikos, Greece)
12 replies
XbenX
May 22, 2019
Null314
36 minutes ago
J
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2021 EGMO P1: {m, 2m+1, 3m} is fantabulous
anser   55
N an hour ago by NicoN9
Source: 2021 EGMO P1
The number 2021 is fantabulous. For any positive integer $m$, if any element of the set $\{m, 2m+1, 3m\}$ is fantabulous, then all the elements are fantabulous. Does it follow that the number $2021^{2021}$ is fantabulous?
55 replies
anser
Apr 13, 2021
NicoN9
an hour ago
J
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Complicated FE
XAN4   0
an hour ago
Source: own
Find all solutions for the functional equation $f(xyz)+\sum_{cyc}f(\frac{yz}x)=f(x)\cdot f(y)\cdot f(z)$, in which $f$: $\mathbb R^+\rightarrow\mathbb R^+$
Note: the solution is actually quite obvious - $f(x)=x^n+\frac1{x^n}$, but the proof is important.
Note 2: it is likely that the result can be generalized into a more advanced questions, potentially involving more bash.
0 replies
XAN4
an hour ago
0 replies
J
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Feet of perpendiculars to diagonal in cyclic quadrilateral
jl_   1
N an hour ago by navier3072
Source: Malaysia IMONST 2 2023 (Primary) P6
Suppose $ABCD$ is a cyclic quadrilateral with $\angle ABC = \angle ADC = 90^{\circ}$. Let $E$ and $F$ be the feet of perpendiculars from $A$ and $C$ to $BD$ respectively. Prove that $BE = DF$.
1 reply
jl_
2 hours ago
navier3072
an hour ago
J
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IMO Shortlist 2014 N5
hajimbrak   58
N an hour ago by Jupiterballs
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.

Proposed by Belgium
58 replies
hajimbrak
Jul 11, 2015
Jupiterballs
an hour ago
J
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Integer a_k such that b - a^n_k is divisible by k
orl   69
N an hour ago by ZZzzyy
Source: IMO Shortlist 2007, N2, Ukrainian TST 2008 Problem 10
Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$.

Author: Dan Brown, Canada
69 replies
orl
Jul 13, 2008
ZZzzyy
an hour ago
J
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interesting function equation (fe) in IR
skellyrah   1
N an hour ago by CrazyInMath
Source: mine
find all function F: IR->IR such that $$ xf(f(y)) + yf(f(x)) = f(xf(y)) + f(xy) $$
1 reply
skellyrah
3 hours ago
CrazyInMath
an hour ago
J
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Find maximum area of right triangle
jl_   1
N 2 hours ago by navier3072
Source: Malaysia IMONST 2 2023 (Primary) P4
Given a right-angled triangle with hypothenuse $2024$, find the maximal area of the triangle.
1 reply
jl_
2 hours ago
navier3072
2 hours ago
J
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Erasing a and b and replacing them with a - b + 1
jl_   1
N 2 hours ago by maromex
Source: Malaysia IMONST 2 2023 (Primary) P5
Ruby writes the numbers $1, 2, 3, . . . , 10$ on the whiteboard. In each move, she selects two distinct numbers, $a$ and $b$, erases them, and replaces them with $a+b-1$. She repeats this process until only one number, $x$, remains. What are all the possible values of $x$?
1 reply
jl_
2 hours ago
maromex
2 hours ago
J
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Prove that sum of 1^3+...+n^3 is a square
jl_   2
N 2 hours ago by NicoN9
Source: Malaysia IMONST 2 2023 (Primary) P1
Prove that for all positive integers $n$, $1^3 + 2^3 + 3^3 +\dots+n^3$ is a perfect square.
2 replies
jl_
3 hours ago
NicoN9
2 hours ago
J
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x^3+y^3 is prime
jl_   2
N 2 hours ago by Jackson0423
Source: Malaysia IMONST 2 2023 (Primary) P3
Find all pairs of positive integers $(x,y)$, so that the number $x^3+y^3$ is a prime.
2 replies
jl_
2 hours ago
Jackson0423
2 hours ago
J
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J
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Inequality for 5 points in the plane
WakeUp   3
N Oct 26, 2019 by Plops
Source: Japanese MO Finals 2000
Given five points $A,B,C,D,E$ in a plane, no three of which are collinear, prove the inequality
\[AB+BC+CA+DE\le AD+AE+BD+BE+CD+CE \]
3 replies
WakeUp
Feb 10, 2011
Plops
Oct 26, 2019
J
Inequality for 5 points in the plane
G H J
Reveal topic tags
inequalities
triangle inequality
geometry proposed
geometry
L
G H BBookmark kLocked kLocked NReply
Source: Japanese MO Finals 2000
The post below has been deleted. Click to close.
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WakeUp
1347 posts
WakeUp
#1 h  Feb 10, 2011, 10:55 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
Given five points $A,B,C,D,E$ in a plane, no three of which are collinear, prove the inequality
\[AB+BC+CA+DE\le AD+AE+BD+BE+CD+CE \]
Z K Y
The post below has been deleted. Click to close.
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SomePig
499 posts
SomePig
#2 h  Feb 12, 2011, 11:20 AM • 2 Y
Y by Adventure10, Mango247
Let F,G,H be the intersections of AD,BE; AD,CE; BD,CE respectively.
AB ≤ AF+BF
BC ≤ BH+CH
CA ≤ CD+DH+GH+FG+EF+AE
DE ≤ DG+EG
AB+BC+CA+DE ≤ (AF+FG+DG)+AE+(BH+DH)+(BF+EF)+CD+(CH+GH+EG)
= AD+AE+BD+BE+CD+CE
Z K Y
The post below has been deleted. Click to close.
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DVDthe1st
341 posts
DVDthe1st
#3 h  Jun 25, 2014, 3:17 PM • 2 Y
Y by mathematiculperson, Adventure10
The above proof is wrong. This is because $D$ might be between $A$ and $G$, and thus $AF+FG+DG$ will be more than $AD$.

Here is my solution:
Z K Y
The post below has been deleted. Click to close.
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Plops
946 posts
Plops
#4 h  Oct 26, 2019, 5:13 AM • 1 Y
Y by Adventure10
I am having trouble applying the triangle inequality to get
DVDthe1st wrote:
$DB+DC\ge AB+AC$
Z K Y
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