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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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Coolabra
Titibuuu   0
17 minutes ago
Source: Mock P1
Let \( a, b, c \) be distinct real numbers such that
\[ a + b + c + \frac{1}{abc} = \frac{19}{2} \]Find the maximum possible value of \( a \).
0 replies
Titibuuu
17 minutes ago
0 replies
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Inspired by Bet667
sqing   5
N 19 minutes ago by sqing
Source: Own
Let $ a,b $ be a real numbers such that $a^3+kab+b^3\ge a^4+b^4.$Prove that
$$1-\sqrt{k+1} \leq a+b\leq 1+\sqrt{k+1} $$Where $ k\geq 0. $
5 replies
sqing
Thursday at 1:03 PM
sqing
19 minutes ago
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Inspired by Kosovo 2010
sqing   2
N 33 minutes ago by sqing
Source: Own
Let $ a,b>0 , a+b\leq k $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq\left(1+\frac{4}{k(k+2)}\right)^2$$$$\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq\left(1+\frac{2}{k+2}\right)^2$$Let $ a,b>0 , a+b\leq 2 $. Prove that
$$\left(1+\frac{1}{a(b+1)}\right)\left(1+\frac{1}{b(a+1)}\right)\geq \frac{9}{4} $$$$\left(1+\frac {a}{b(a+1)}\right)\left(1+\frac {b}{a(b+1)}\right) \geq \frac{9}{4} $$
2 replies
sqing
Yesterday at 3:56 AM
sqing
33 minutes ago
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Find all real numbers
sqing   6
N 34 minutes ago by sqing
Source: IMOC 2021 A1
Find all real numbers x that satisfies$$\sqrt{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}+\sqrt{1-\frac{1}{\sqrt{x-\frac{1}{x}}+\sqrt{1-\frac{1}{x}}}}=x.$$2021 IMOC Problems
6 replies
sqing
Aug 11, 2021
sqing
34 minutes ago
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Inequalities
sqing   7
N 36 minutes ago by sqing
Let $ a,b>0, a^2+ab+b^2 \geq 6 $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$a^4+ab+b^4 \leq 10$$Let $ a,b>0, a^2+ab+b^2 \geq \frac{15}{2} $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$-\frac{1}{8}\leq a^4-ab+b^4\leq 10$$
7 replies
sqing
Thursday at 2:42 PM
sqing
36 minutes ago
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IMO 2016 Problem 4
termas   54
N an hour ago by OronSH
Source: IMO 2016 (day 2)
A set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$is fragrant?
54 replies
termas
Jul 12, 2016
OronSH
an hour ago
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Cyclic equality implies equal sum of squares
blackbluecar   33
N 2 hours ago by Vivaandax
Source: 2021 Iberoamerican Mathematical Olympiad, P4
Let $a,b,c,x,y,z$ be real numbers such that

\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]
Show that $a^2+b^2+c^2=x^2+y^2+z^2$.
33 replies
blackbluecar
Oct 21, 2021
Vivaandax
2 hours ago
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Compilation of functions problems
Saucepan_man02   2
N 2 hours ago by Saucepan_man02
Could anyone post some handout/compilation of problems related to functions (difficulty similar to AIME/ARML/HMMT etc)?

Thanks..
2 replies
Saucepan_man02
May 7, 2025
Saucepan_man02
2 hours ago
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How many triangles
Ecrin_eren   5
N 2 hours ago by jasperE3


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


5 replies
Ecrin_eren
May 2, 2025
jasperE3
2 hours ago
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Proving that the line passes through the midpoint.
MTA_2024   1
N 3 hours ago by MTA_2024
Let $ABC$ be a triangle of orthocenter $H$. The circle of diameter $AC$ and the circumcircle of triangle $AHB$ intersect a second time in $K$.
Prove that the line $(CK)$ passes through the midpoint of segment $HB$.
1 reply
MTA_2024
May 7, 2025
MTA_2024
3 hours ago
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Interesting functional equation with geometry
User21837561   3
N Yesterday at 6:34 PM by Double07
Source: BMOSL 2024 G7
For an acute triangle $ABC$, let $O$ be the circumcentre, $H$ be the orthocentre, and $G$ be the centroid.
Let $f:\pi\rightarrow\mathbb R$ satisfy the following condition:
$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$
Prove that $f$ is constant.
3 replies
User21837561
Yesterday at 8:14 AM
Double07
Yesterday at 6:34 PM
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greatest volume
hzbrl   1
N Yesterday at 6:32 PM by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
1 reply
hzbrl
Thursday at 9:56 AM
hzbrl
Yesterday at 6:32 PM
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IMO 2010 Problem 3
canada   59
N Yesterday at 6:23 PM by pi271828
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$

Proposed by Gabriel Carroll, USA
59 replies
canada
Jul 7, 2010
pi271828
Yesterday at 6:23 PM
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Functional equation
Nima Ahmadi Pour   100
N Yesterday at 5:11 PM by jasperE3
Source: ISl 2005, A2, Iran prepration exam
We denote by $\mathbb{R}^+$ the set of all positive real numbers.

Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:
\[f(x)f(y)=2f(x+yf(x))\]
for all positive real numbers $x$ and $y$.

Proposed by Nikolai Nikolov, Bulgaria
100 replies
Nima Ahmadi Pour
Apr 24, 2006
jasperE3
Yesterday at 5:11 PM
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a point P such that AP +BP+CP is minimised
Iris Aliaj   3
N Jul 22, 2004 by beta
Triangle ABC has a right angle at A. Among all the points P on the perimeter of the triangle, find the position of P such that
AP + BP + CP
is minimised
3 replies
Iris Aliaj
Jul 22, 2004
beta
Jul 22, 2004
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a point P such that AP +BP+CP is minimised
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Reveal topic tags
geometry
perimeter
quadratics
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Iris Aliaj
165 posts
Iris Aliaj
#1 h  Jul 22, 2004, 12:15 PM • 2 Y
Y by Adventure10, Mango247
Triangle ABC has a right angle at A. Among all the points P on the perimeter of the triangle, find the position of P such that
AP + BP + CP
is minimised
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beta
3001 posts
beta
#2 h  Jul 22, 2004, 8:02 PM • 2 Y
Y by Adventure10, Mango247
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This post has been edited 2 times. Last edited by beta, Jul 28, 2004, 8:26 PM
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Pork_Chop8
179 posts
Pork_Chop8
#3 h  Jul 22, 2004, 9:08 PM • 2 Y
Y by Adventure10, Mango247
Edit:
The minimum of a quadratic $f(x) = ax^{2}+bx+c$ with $a>0$, occurs when $x=\displaystyle-\frac{b}{2a}$, which is not, of course, the same as when $x=\displaystyle-\frac{b}{a}$. So, in conclusion, thank you beta for the mistake check. Our methods were exactly the same.
This post has been edited 1 time. Last edited by Pork_Chop8, Jul 23, 2004, 3:06 AM
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3001 posts
beta
#4 h  Jul 22, 2004, 10:21 PM • 2 Y
Y by Adventure10, Mango247
Pork_Chop8 wrote:
$\displaystyle \frac{\sqrt{(c^{2}+1)k^{2}-2kc+c^{2}}}{c}$ which is minimized when $\displaystyle k=-\frac{-2c}{c^{2}+1}$ and plugging that in for k, we get $\frac{\sqrt{c^{2}}}{c}=1$. So the minimum would be $BC + 1$ or $BC + AB$.
I am pretty sure if you minimize $\displaystyle \frac{\sqrt{(c^{2}+1)k^{2}-2kc+c^{2}}}{c}$ right, it is $\frac{c}{\sqrt{c^2+1}}$, not 1, because if AP is minimized, then AP is perpendicular to BC, which according to your definition, is $\frac{c}{\sqrt{c^2+1}}$, there's no way AB is perpendicular to BC..
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