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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
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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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\frac{1}{5-2a}
Havu   1
N 2 hours ago by Havu
Let $a,b,c \ge \frac{1}{2}$ and $a^2+b^2+c^2=3$. Find minimum:
\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]
1 reply
Havu
Yesterday at 9:56 AM
Havu
2 hours ago
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<DPA+ <AQD =< QIP wanted, incircle circumcircle related
parmenides51   41
N 3 hours ago by Ilikeminecraft
Source: IMo 2019 SL G6
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.

(Slovakia)
41 replies
parmenides51
Sep 22, 2020
Ilikeminecraft
3 hours ago
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Pretty hard functional equation
vralex   5
N 3 hours ago by jasperE3
Source: National MO, 9th grade
Find all injective functions $ f:\mathbb{Z} \rightarrow \mathbb{Z} $ so that for every $n$ in $\mathbb{Z} , f (f (n))-f(n)-1=0$.
5 replies
vralex
Apr 29, 2020
jasperE3
3 hours ago
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Parallelity and equal angles given, wanted an angle equality
BarisKoyuncu   5
N 3 hours ago by SleepyGirraffe
Source: 2022 Turkey JBMO TST P4
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
5 replies
BarisKoyuncu
Mar 15, 2022
SleepyGirraffe
3 hours ago
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Cyclic points and concurrency [1st Lemoine circle]
shobber   10
N 4 hours ago by Ilikeminecraft
Source: China TST 2005
Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$.
(1) Prove that $F,B,C,E$ are concyclic.

(2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.
10 replies
shobber
Jun 27, 2006
Ilikeminecraft
4 hours ago
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Hard functional equation
Jessey   4
N 4 hours ago by jasperE3
Source: Belarus 2005
Find all functions $f:N -$> $N$ that satisfy $f(m-n+f(n)) = f(m)+f(n)$, for all $m, n$ € $N$.
4 replies
Jessey
Mar 11, 2020
jasperE3
4 hours ago
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Vertices of a convex polygon if and only if m(S) = f(n)
orl   12
N 5 hours ago by Maximilian113
Source: IMO Shortlist 2000, C3
Let $ n \geq 4$ be a fixed positive integer. Given a set $ S = \{P_1, P_2, \ldots, P_n\}$ of $ n$ points in the plane such that no three are collinear and no four concyclic, let $ a_t,$ $ 1 \leq t \leq n,$ be the number of circles $ P_iP_jP_k$ that contain $ P_t$ in their interior, and let \[m(S)=a_1+a_2+\cdots + a_n.\]Prove that there exists a positive integer $ f(n),$ depending only on $ n,$ such that the points of $ S$ are the vertices of a convex polygon if and only if $ m(S) = f(n).$
12 replies
orl
Aug 10, 2008
Maximilian113
5 hours ago
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Imo Shortlist Problem
Lopes   35
N 5 hours ago by Maximilian113
Source: IMO Shortlist 2000, Problem N4
Find all triplets of positive integers $ (a,m,n)$ such that $ a^m + 1 \mid (a + 1)^n$.
35 replies
Lopes
Feb 27, 2005
Maximilian113
5 hours ago
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Inspired by Humberto_Filho
sqing   0
5 hours ago
Source: Own
Let $ a,b\geq 0 $ and $a + b \leq 2$. Prove that
$$\frac{a^2+1}{(( a+ b)^2+1)^2} \geq \frac{1}{25} $$$$\frac{(a^2+1)(b^2+1)}{((a+b)^2+1)^2} \geq \frac{4}{25} $$$$ \frac{a^2+1}{(( a+ 2b)^2+1)^2} \geq \frac{1}{289} $$$$ \frac{a^2+1}{((2a+ b)^2+1)^2} \geq \frac{5}{289} $$


0 replies
sqing
5 hours ago
0 replies
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Inequalities
Scientist10   2
N 5 hours ago by arqady
If $x, y, z \in \mathbb{R}$, then prove that the following inequality holds:
\[ \sum_{\text{cyc}} \sqrt{1 + \left(x\sqrt{1 + y^2} + y\sqrt{1 + x^2}\right)^2} \geq \sum_{\text{cyc}} xy + 2\sum_{\text{cyc}} x \]
2 replies
Scientist10
Yesterday at 6:36 PM
arqady
5 hours ago
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$n$ with $2000$ divisors divides $2^n+1$ (IMO 2000)
Valentin Vornicu   65
N 5 hours ago by ray66
Source: IMO 2000, Problem 5, IMO Shortlist 2000, Problem N3
Does there exist a positive integer $ n$ such that $ n$ has exactly 2000 prime divisors and $ n$ divides $ 2^n + 1$?
65 replies
Valentin Vornicu
Oct 24, 2005
ray66
5 hours ago
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Find the smallest of sum of elements
hlminh   0
5 hours ago
Let $S=\{1,2,...,2014\}$ and $X=\{a_1,a_2,...,a_{30}\}$ is a subset of $S$ such that if $a,b\in X,a+b\leq 2014$ then $a+b\in X.$ Find the smallest of $\dfrac{a_1+a_2+\cdots+a_{30}}{30}.$
0 replies
hlminh
5 hours ago
0 replies
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Easy IMO 2023 NT
799786   133
N 5 hours ago by Maximilian113
Source: IMO 2023 P1
Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.
133 replies
799786
Jul 8, 2023
Maximilian113
5 hours ago
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Complicated FE
XAN4   2
N 6 hours ago by cazanova19921
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.
2 replies
XAN4
Yesterday at 11:53 AM
cazanova19921
6 hours ago
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Flanders 1
Arne   2
N Sep 7, 2003 by Arne
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
2 replies
Arne
Sep 7, 2003
Arne
Sep 7, 2003
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Flanders 1
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Reveal topic tags
ratio
geometry
vector
geometry solved
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Arne
3660 posts
Arne
#1 h  Sep 7, 2003, 9:41 AM • 2 Y
Y by Adventure10, Mango247
Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.
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grobber
7849 posts
grobber
#2 h  Sep 7, 2003, 10:58 AM • 2 Y
Y by Adventure10, Mango247
The ratio is 3/4; we just construct the triangle having the medians as sides by formung some parallelograms. This is possible because the sum of the vectors AM, BN, CP (where M, N, P are midpts of BC, CA, AB) is 0, so there is a triangle having sides equal to AM, BN, CP and parallel to these.
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Arne
3660 posts
Arne
#3 h  Sep 7, 2003, 11:00 AM • 2 Y
Y by Adventure10, Mango247
Indeed Grobber !
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