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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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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
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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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Question on Balkan SL
Fmimch   4
N 10 minutes ago by BreezeCrowd
Does anyone know where to find the Balkan MO Shortlist 2024? If you have the file, could you send in this thread? Thank you!
4 replies
+1 w
Fmimch
Apr 30, 2025
BreezeCrowd
10 minutes ago
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(3^{p-1} - 1)/p is a perfect square for prime p
parmenides51   4
N 22 minutes ago by Rayvhs
Source: 2017 Saudi Arabia JBMO TST 1.2
Find all prime numbers $p$ such that $\frac{3^{p-1} - 1}{p}$ is a perfect square.
4 replies
+1 w
parmenides51
May 28, 2020
Rayvhs
22 minutes ago
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Rays, incircle, angles...
mathisreal   3
N 38 minutes ago by Assassino9931
Source: Rioplatense L-3 2022 #4
Let $ABC$ be a triangle with incenter $I$. Let $D$ be the point of intersection between the incircle and the side $BC$, the points $P$ and $Q$ are in the rays $IB$ and $IC$, respectively, such that $\angle IAP=\angle CAD$ and $\angle IAQ=\angle BAD$. Prove that $AP=AQ$.
3 replies
mathisreal
Dec 13, 2022
Assassino9931
38 minutes ago
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Find the value
sqing   0
an hour ago
Source: Own
Let $ a,b $ be real numbers such that $ (a^2 + b^2) (a + 1) (b + 1) = a ^ 3 + b ^ 3 =2 $. Find the value of $ a b .$

Let $ a,b $ be real numbers such that $ (a^2 + b^2) (a + 1) (b + 1) = 2 $ and $ a ^ 3 + b ^ 3 = 1 $. Find the value of $ a + b .$
0 replies
sqing
an hour ago
0 replies
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Wordy Geometry in Taiwan TST
ckliao914   9
N an hour ago by Scilyse
Source: 2023 Taiwan TST Round 3 Mock Exam 6
Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.
9 replies
ckliao914
Apr 29, 2023
Scilyse
an hour ago
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Factorial Divisibility
Aryan-23   47
N an hour ago by ezpotd
Source: IMO SL 2022 N2
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
47 replies
Aryan-23
Jul 9, 2023
ezpotd
an hour ago
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2-var inequality
sqing   3
N an hour ago by sqing
Source: Own
Let $ a,b> 0 ,a^3+ab+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq 8$$$$ (a^2+b^2)(a+1)(b+1) \leq 8$$Let $ a,b> 0 ,a^3+ab(a+b)+b^3=3.$ Prove that
$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$
3 replies
sqing
2 hours ago
sqing
an hour ago
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Infinite number of sets with an intersection property
Drytime   8
N 2 hours ago by math90
Source: Romania TST 2013 Test 2 Problem 4
Let $k$ be a positive integer larger than $1$. Build an infinite set $\mathcal{A}$ of subsets of $\mathbb{N}$ having the following properties:

(a) any $k$ distinct sets of $\mathcal{A}$ have exactly one common element;
(b) any $k+1$ distinct sets of $\mathcal{A}$ have void intersection.
8 replies
Drytime
Apr 26, 2013
math90
2 hours ago
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Factorials divide
va2010   37
N 2 hours ago by ND_
Source: 2015 ISL N2
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.
37 replies
va2010
Jul 7, 2016
ND_
2 hours ago
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IMO Shortlist 2011, Number Theory 2
orl   24
N 2 hours ago by ezpotd
Source: IMO Shortlist 2011, Number Theory 2
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.

Proposed by Luxembourg
24 replies
orl
Jul 11, 2012
ezpotd
2 hours ago
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Inequality in triangle
Nguyenhuyen_AG   3
N 2 hours ago by Nguyenhuyen_AG
Let $a,b,c$ be the lengths of the sides of a triangle. Prove that
\[\frac{1}{(a-4b)^2}+\frac{1}{(b-4c)^2}+\frac{1}{(c-4a)^2} \geqslant \frac{1}{ab+bc+ca}.\]
3 replies
Nguyenhuyen_AG
Today at 6:17 AM
Nguyenhuyen_AG
2 hours ago
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Problem 1
randomusername   73
N 3 hours ago by ND_
Source: IMO 2015, Problem 1
We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.

(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Proposed by Netherlands
73 replies
randomusername
Jul 10, 2015
ND_
3 hours ago
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x is rational implies y is rational
pohoatza   44
N 3 hours ago by ezpotd
Source: IMO Shortlist 2006, N2, VAIMO 2007, Problem 6
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.

Proposed by J.P. Grossman, Canada
44 replies
pohoatza
Jun 28, 2007
ezpotd
3 hours ago
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Multiplicative function
Tales   37
N 3 hours ago by ezpotd
Source: IMO Shortlist 2004, number theory problem 2
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.

b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.

c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
37 replies
Tales
Mar 23, 2005
ezpotd
3 hours ago
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Interesting inequality
martingraf   2
N Nov 9, 2010 by kuing
a,b real numbers(a>0 , b>0) and n $ \in \mathbb{Z} $ , n $ \ge $ 1.
Prove that:$ \frac{1}{a+b} $+$ \frac{1}{a+2b} $+...+$ \frac{1}{a+nb} $ < $ \frac{n}{\sqrt{a(a+nb)}} $.
2 replies
martingraf
Nov 9, 2010
kuing
Nov 9, 2010
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Interesting inequality
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Reveal topic tags
inequalities
inequalities proposed
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martingraf
94 posts
martingraf
#1 h  Nov 9, 2010, 3:01 PM • 1 Y
Y by Adventure10
a,b real numbers(a>0 , b>0) and n $ \in \mathbb{Z} $ , n $ \ge $ 1.
Prove that:$ \frac{1}{a+b} $+$ \frac{1}{a+2b} $+...+$ \frac{1}{a+nb} $ < $ \frac{n}{\sqrt{a(a+nb)}} $.
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mavropnevma
15142 posts
mavropnevma
#2 h  Nov 9, 2010, 3:44 PM • 2 Y
Y by Adventure10, Mango247
See http://www.artofproblemsolving.com/Forum/viewtopic.php?f=36&t=375644&hilit=%22a+nb%22
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kuing
1008 posts
kuing
#3 h  Nov 9, 2010, 4:00 PM • 2 Y
Y by Adventure10, Mango247
martingraf wrote:
a,b real numbers(a>0 , b>0) and n $ \in \mathbb{Z} $ , n $ \ge $ 1.
Prove that:$ \frac{1}{a+b} $+$ \frac{1}{a+2b} $+...+$ \frac{1}{a+nb} $ < $ \frac{n}{\sqrt{a(a+nb)}} $.
by CS
$\left(\frac{1}{a+b}+\frac{1}{a+2b}+\cdots+\frac{1}{a+nb}\right)^2$

$< n\left(\frac{1}{(a+b)^2}+\frac{1}{(a+2b)^2}+\cdots+\frac{1}{(a+nb)^2}\right)$

$<n\left(\frac{1}{(a+\frac{b}{2})(a+\frac{3b}{2})}+\frac{1}{(a+\frac{3b}{2})(a+\frac{5b}{2})}+\cdots+\frac{1}{(a+\frac{2n-1}{2}b)(a+\frac{2n+1}{2}b)}\right)$

$=\frac{n}{b}\left(\frac{1}{a+\frac{b}{2}}-\frac{1}{a+\frac{3b}{2}}+\frac{1}{a+\frac{3b}{2}}-\frac{1}{a+\frac{5b}{2}}+\cdots+\frac{1}{a+\frac{2n-1}{2}b}-\frac{1}{a+\frac{2n+1}{2}b}\right)$

$=\frac{n}{b}\left(\frac{1}{a+\frac{b}{2}}-\frac{1}{a+\frac{2n+1}{2}b}\right)$

$=\frac{n^2}{(a+\frac{b}{2})(a+\frac{2n+1}{2}b)}$

$<\frac{n^2}{a(a+nb)}$

done.
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