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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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combi/nt
blug   0
6 minutes ago
Prove that every positive integer $n$ can be written in the form
$$n=2^{a_1}3^{b_1}+2^{a_2}3^{b_2}+..., $$where $a_i, b_j$ are non negative integers, such that
$$2^x3^y\nmid 2^z3^t$$for every $x, y, z, t$.
0 replies
blug
6 minutes ago
0 replies
J
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Interesting inequalities
sqing   2
N 13 minutes ago by sqing
Source: Own
Let $ a,b,c\geq 0 , a(b+c)=k.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{4\sqrt{k}-6}{ k-2}$$Where $5\leq k\in N^+.$
Let $ a,b,c\geq 0 , a(b+c)=9.$ Prove that
$$\frac{1}{a+1}+\frac{2}{b+1}+\frac{1}{c+1}\geq \frac{6}{7}$$
2 replies
1 viewing
sqing
2 hours ago
sqing
13 minutes ago
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Find smallest value of (x^2 + y^2 + z^2)/(xyz)
orl   7
N 22 minutes ago by Bryan0224
Source: CWMO 2001, Problem 4
Let $ x, y, z$ be real numbers such that $ x + y + z \geq xyz$. Find the smallest possible value of $ \frac {x^2 + y^2 + z^2}{xyz}$.
7 replies
orl
Dec 27, 2008
Bryan0224
22 minutes ago
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easy substitutions for a functional in reals
Circumcircle   9
N 44 minutes ago by Bardia7003
Source: Kosovo Math Olympiad 2025, Grade 11, Problem 2
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ with the property that for every real numbers $x$ and $y$ it holds that
$$f(x+yf(x+y))=f(x)+f(xy)+y^2.$$
9 replies
Circumcircle
Nov 16, 2024
Bardia7003
44 minutes ago
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writing words around circle, two letters
jasperE3   1
N an hour ago by pi_quadrat_sechstel
Source: VJIMC 2000 2.2
If we write the sequence $\text{AAABABBB}$ along the perimeter of a circle, then every word of the length $3$ consisting of letters $A$ and $B$ (i.e. $\text{AAA}$, $\text{AAB}$, $\text{ABA}$, $\text{BAB}$, $\text{ABB}$, $\text{BBB}$, $\text{BBA}$, $\text{BAA}$) occurs exactly once on the perimeter. Decide whether it is possible to write a sequence of letters from a $k$-element alphabet along the perimeter of a circle in such a way that every word of the length $l$ (i.e. an ordered $l$-tuple of letters) occurs exactly once on the perimeter.
1 reply
jasperE3
Jul 27, 2021
pi_quadrat_sechstel
an hour ago
J
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Interesting inequality
imnotgoodatmathsorry   0
an hour ago
Let $x,y,z > \frac{1}{2}$ and $x+y+z=3$.Prove that:
$\sqrt{x^3+y^3+3xy-1}+\sqrt{y^3+z^3+3yz-1}+\sqrt{z^3+x^3+3zx-1}+\frac{1}{4}(x+5)(y+5)(z+5) \le 60$
0 replies
imnotgoodatmathsorry
an hour ago
0 replies
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Arithmetic Sequence of Products
GrantStar   19
N an hour ago by OronSH
Source: IMO Shortlist 2023 N4
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
19 replies
GrantStar
Jul 17, 2024
OronSH
an hour ago
J
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Inequality Involving Complex Numbers with Modulus Less Than 1
tom-nowy   0
an hour ago
Let $x,y,z$ be complex numbers such that $|x|<1, |y|<1,$ and $|z|<1$.
Prove that $$ |x+y+z|^2 +3>|xy+yz+zx|^2+3|xyz|^2 .$$
0 replies
tom-nowy
an hour ago
0 replies
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Inequality
nguyentlauv   2
N an hour ago by nguyentlauv
Source: Own
Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$, prove that
$$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$$
2 replies
nguyentlauv
May 6, 2025
nguyentlauv
an hour ago
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japan 2021 mo
parkjungmin   0
an hour ago

The square box question

Is there anyone who can release it
0 replies
parkjungmin
an hour ago
0 replies
J
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easy sequence
Seungjun_Lee   17
N an hour ago by GreekIdiot
Source: KMO 2023 P1
A sequence of positive reals $\{ a_n \}$ is defined below. $$a_0 = 1, a_1 = 3, a_{n+2} = \frac{a_{n+1}^2+2}{a_n}$$Show that for all nonnegative integer $n$, $a_n$ is a positive integer.
17 replies
Seungjun_Lee
Nov 4, 2023
GreekIdiot
an hour ago
J
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Japan MO Finals 2023
parkjungmin   0
an hour ago
It's hard. Help me
0 replies
parkjungmin
an hour ago
0 replies
J
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I Brazilian TST 2007 - Problem 4
e.lopes   77
N an hour ago by alexanderhamilton124
Source: 2007 Brazil TST, Russia TST, and AIMO; also SL 2006 N5
Find all integer solutions of the equation \[\frac {x^{7} - 1}{x - 1} = y^{5} - 1.\]
77 replies
e.lopes
Mar 11, 2007
alexanderhamilton124
an hour ago
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Japan MO Finals 2024
parkjungmin   0
an hour ago
Source: Please tell me the question
Please tell me the question
0 replies
parkjungmin
an hour ago
0 replies
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J
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Indian Mathematical Olympiad 1987 - Problem 7
Leon   4
N Nov 5, 2024 by namanrobin08
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
4 replies
Leon
Aug 13, 2009
namanrobin08
Nov 5, 2024
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Indian Mathematical Olympiad 1987 - Problem 7
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Reveal topic tags
geometry
ratio
algebra proposed
algebra
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Leon
256 posts
Leon
#1 h  Aug 13, 2009, 12:15 PM • 2 Y
Y by Adventure10, Mango247
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.
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livetolove212
859 posts
livetolove212
#2 h  Aug 14, 2009, 2:21 PM • 2 Y
Y by Adventure10, Mango247
This topic must be in geometry forum!
Given $ AH$ such that $ AH = h_a$.
Construct a line d which passes through H and perpendicular to $ AH. (A, m_a)\cap d = \{M\}$
Since $ ah_a = bh_b$ then $ k = \frac {h_a}{h_b} = \frac {b}{a} = \frac {AC}{BC}$
Then $ \frac {AC}{MC} = 2k$.
Construct Apollonius circle of a line segment AM wrt ratio 2k, it cuts d at C. B is a point such that $ MC = MB = 1/2BC$
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Γιώργος
512 posts
Γιώργος
#3 h  Aug 17, 2009, 3:40 PM • 2 Y
Y by Adventure10, Mango247
Leon wrote:
Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.

I think, that is I.M.O 1960 :wink:
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namanrobin08
99 posts
namanrobin08
#4 h  Nov 5, 2024, 5:15 PM
Y by
Its IMO 1960 P4 :spidy:
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namanrobin08
99 posts
namanrobin08
#5 h  Nov 5, 2024, 5:26 PM
Y by
solution
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