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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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subsets of {1,2,...,mn}
N.T.TUAN   10
N 6 minutes ago by de-Kirschbaum
Source: USA TST 2005, Problem 1
Let $n$ be an integer greater than $1$. For a positive integer $m$, let $S_{m}= \{ 1,2,\ldots, mn\}$. Suppose that there exists a $2n$-element set $T$ such that
(a) each element of $T$ is an $m$-element subset of $S_{m}$;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$.

Determine the maximum possible value of $m$ in terms of $n$.
10 replies
N.T.TUAN
May 14, 2007
de-Kirschbaum
6 minutes ago
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Sum and product of digits
Sadigly   4
N 9 minutes ago by jasperE3
Source: Azerbaijan NMO 2018
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
4 replies
Sadigly
Sunday at 9:19 PM
jasperE3
9 minutes ago
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Geometry
smartvong   0
13 minutes ago
Source: UM Mathematical Olympiad 2024
Let $P$ be a point inside a triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2, C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
0 replies
smartvong
13 minutes ago
0 replies
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angles in triangle
AndrewTom   34
N 30 minutes ago by happypi31415
Source: BrMO 2012/13 Round 2
The point $P$ lies inside triangle $ABC$ so that $\angle ABP = \angle PCA$. The point $Q$ is such that $PBQC$ is a parallelogram. Prove that $\angle QAB = \angle CAP$.
34 replies
AndrewTom
Feb 1, 2013
happypi31415
30 minutes ago
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Hard to approach it !
BogG   130
N 2 hours ago by Ilikeminecraft
Source: Swiss Imo Selection 2006
Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.
130 replies
BogG
May 25, 2006
Ilikeminecraft
2 hours ago
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A game with balls and boxes
egxa   5
N 2 hours ago by compoly2010
Source: Turkey JBMO TST 2023 Day 1 P4
Initially, Aslı distributes $1000$ balls to $30$ boxes as she wishes. After that, Aslı and Zehra make alternated moves which consists of taking a ball in any wanted box starting with Aslı. One who takes the last ball from any box takes that box to herself. What is the maximum number of boxes can Aslı guarantee to take herself regardless of Zehra's moves?
5 replies
egxa
Apr 30, 2023
compoly2010
2 hours ago
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Two lines meet on semicircle
va2010   87
N 2 hours ago by Ilikeminecraft
Source: 2015 ISL G3
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
87 replies
va2010
Jul 7, 2016
Ilikeminecraft
2 hours ago
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something...
SunnyEvan   2
N 3 hours ago by SunnyEvan
Source: unknown
Try to prove : $$ \sum csc^{20} \frac{2^{i} \pi}{7} csc^{23} \frac{2^{j}\pi }{7} csc^{2023} \frac{2^{k} \pi}{7} $$is a rational number.
Where $ (i,j,k)=(1,2,3) $ and other permutations.
2 replies
SunnyEvan
May 5, 2025
SunnyEvan
3 hours ago
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USAMO 2002 Problem 2
MithsApprentice   34
N 3 hours ago by Giant_PT
Let $ABC$ be a triangle such that
\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]
where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.
34 replies
MithsApprentice
Sep 30, 2005
Giant_PT
3 hours ago
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Another config geo with concurrent lines
a_507_bc   17
N 3 hours ago by Rayvhs
Source: BMO SL 2023 G5
Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.
17 replies
a_507_bc
May 3, 2024
Rayvhs
3 hours ago
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Nice sequence problem.
mathlover1231   1
N 4 hours ago by vgtcross
Source: Own
Scientists found a new species of bird called “N-coloured rainbow”. They also found out 3 interesting facts about the bird’s life: 1) every day, N-coloured rainbow is coloured in one of N colors.
2) every day, the color is different from yesterday (not every previous day, just yesterday).
3) there are no four days i, j, k, l in the bird’s life such that i<j<k<l with colours a, b, c, d respectively for which a=c ≠ b=d.
Find the greatest possible age (in days) of this bird as a function of N.
1 reply
mathlover1231
Apr 10, 2025
vgtcross
4 hours ago
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Three circles are concurrent
Twoisaprime   23
N 4 hours ago by Curious_Droid
Source: RMM 2025 P5
Let triangle $ABC$ be an acute triangle with $AB<AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. Let $\Gamma$ be the circle $BOC$. The line $AO$ and the circle of radius $AO$ centered at $A$ cross $\Gamma$ at $A’$ and $F$, respectively. Prove that $\Gamma$ , the circle on diameter $AA’$ and circle $AFH$ are concurrent.
Proposed by Romania, Radu-Andrew Lecoiu
23 replies
Twoisaprime
Feb 13, 2025
Curious_Droid
4 hours ago
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|a_i/a_j - a_k/a_l| <= C
mathwizard888   32
N 4 hours ago by ezpotd
Source: 2016 IMO Shortlist A2
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
32 replies
mathwizard888
Jul 19, 2017
ezpotd
4 hours ago
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Two lines meeting on circumcircle
Zhero   54
N 4 hours ago by Ilikeminecraft
Source: ELMO Shortlist 2010, G4; also ELMO #6
Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$.

Amol Aggarwal.
54 replies
Zhero
Jul 5, 2012
Ilikeminecraft
4 hours ago
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Not easy
izat   4
N Mar 5, 2016 by Abubakir
FInd all $ f : \mathbb{R}\to\mathbb{R}$ such that $f(x^2+y+f(y))=2y+f(x)^2$. For all $x, y$
4 replies
izat
Jan 28, 2016
Abubakir
Mar 5, 2016
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Not easy
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Reveal topic tags
function
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izat
77 posts
izat
#1 h  Jan 28, 2016, 4:41 AM • 1 Y
Y by Adventure10
FInd all $ f : \mathbb{R}\to\mathbb{R}$ such that $f(x^2+y+f(y))=2y+f(x)^2$. For all $x, y$
Z K Y
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Tintarn
9042 posts
Tintarn
#2 h  Jan 28, 2016, 8:03 PM • 1 Y
Y by Adventure10
This is an old problem. But since I just can't find it I will present the standard solution:
Solution
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KMI
14 posts
KMI
#3 h  Mar 5, 2016, 5:00 AM • 1 Y
Y by Adventure10
Tintarn wrote:
we obtain $a^2=b^2$. So $a^2=b^2 \Leftrightarrow f(a)^2=f(b)^2$.
Since $f$ is surjective this implies $f(-x)=-f(x)$ for all $x$.

for surjective function ? not injective?
typo or error?
Z K Y
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don2001
170 posts
don2001
#4 h  Mar 5, 2016, 5:25 AM • 2 Y
Y by Adventure10, Mango247
Another link of problem http://www.artofproblemsolving.com/community/c6t298f6h1207079_functional_eq
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Abubakir
68 posts
Abubakir
#5 h  Mar 5, 2016, 6:50 AM • 2 Y
Y by Tintarn, Adventure10
KMI wrote:
Tintarn wrote:
we obtain $a^2=b^2$. So $a^2=b^2 \Leftrightarrow f(a)^2=f(b)^2$.
Since $f$ is surjective this implies $f(-x)=-f(x)$ for all $x$.

for surjective function ? not injective?
typo or error?

Maybe Tintarn meant that, for all $x$ positive $f(x)$ is positive. $f$ is surjective, so we have that there is $y$ such that $f(y)=-f(x)$. We know that $y$ could be only $-x$ or $x$. The latter is wrong, so $y=-x$.
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