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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
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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
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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Special line through antipodal
Phorphyrion   9
N 11 minutes ago by ihategeo_1969
Source: 2025 Israel TST Test 1 P2
Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.
9 replies
Phorphyrion
Oct 28, 2024
ihategeo_1969
11 minutes ago
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Triangle form by perpendicular bisector
psi241   50
N an hour ago by Ilikeminecraft
Source: IMO Shortlist 2018 G5
Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. A line $\ell$ intersects the lines $AI$, $BI$, and $CI$ at points $D$, $E$, and $F$, respectively, distinct from the points $A$, $B$, $C$, and $I$. The perpendicular bisectors $x$, $y$, and $z$ of the segments $AD$, $BE$, and $CF$, respectively determine a triangle $\Theta$. Show that the circumcircle of the triangle $\Theta$ is tangent to $\Omega$.
50 replies
psi241
Jul 17, 2019
Ilikeminecraft
an hour ago
J
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Sequence with infinite primes which we see again and again and again
Assassino9931   3
N an hour ago by grupyorum
Source: Balkan MO Shortlist 2024 N6
Let $c$ be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence $a_1 = c$, $a_{n+1} = a_n^3 + c$.
3 replies
Assassino9931
Apr 27, 2025
grupyorum
an hour ago
J
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Integer roots preserved under linear function of polynomial
alifenix-   23
N an hour ago by Mathandski
Source: USEMO 2019/2
Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials)
such that
[list]
[*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$;
[*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does.
[/list]

Carl Schildkraut
23 replies
alifenix-
May 23, 2020
Mathandski
an hour ago
J
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BMO 2024 SL A3
MuradSafarli   5
N an hour ago by Nuran2010

A3.
Find all triples \((a, b, c)\) of positive real numbers that satisfy the system:
\[ \begin{aligned} 11bc - 36b - 15c &= abc \\ 12ca - 10c - 28a &= abc \\ 13ab - 21a - 6b &= abc. \end{aligned} \]
5 replies
MuradSafarli
Apr 27, 2025
Nuran2010
an hour ago
J
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Cool functional equation
Rayanelba   4
N 2 hours ago by ATM_
Source: Own
Find all functions $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ that verify the following equation for all $x,y\in \mathbb{Z}_{>0}$:
$max(f^{f(y)}(x),f^{f(y)}(y))|min(x,y)$
4 replies
Rayanelba
4 hours ago
ATM_
2 hours ago
J
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primes,exponentials,factorials
skellyrah   3
N 2 hours ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
3 replies
skellyrah
5 hours ago
skellyrah
2 hours ago
J
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af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
shoki   26
N 2 hours ago by MathLuis
Source: Iran TST 2011 - Day 4 - Problem 3
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
26 replies
shoki
May 14, 2011
MathLuis
2 hours ago
J
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Very easy NT
GreekIdiot   8
N 2 hours ago by vsamc
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
8 replies
GreekIdiot
Today at 2:49 PM
vsamc
2 hours ago
J
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Another quadrilateral in a circle
v_Enhance   110
N 2 hours ago by Marco22
Source: APMO 2013, Problem 5
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
110 replies
v_Enhance
May 3, 2013
Marco22
2 hours ago
J
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Rectangle EFGH in incircle, prove that QIM = 90
v_Enhance   64
N 3 hours ago by lpieleanu
Source: Taiwan 2014 TST1, Problem 3
Let $ABC$ be a triangle with incenter $I$, and suppose the incircle is tangent to $CA$ and $AB$ at $E$ and $F$. Denote by $G$ and $H$ the reflections of $E$ and $F$ over $I$. Let $Q$ be the intersection of $BC$ with $GH$, and let $M$ be the midpoint of $BC$. Prove that $IQ$ and $IM$ are perpendicular.
64 replies
v_Enhance
Jul 18, 2014
lpieleanu
3 hours ago
J
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Queue geo
vincentwant   2
N 3 hours ago by MathLuis
Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $Y, Z$ be the feet of the altitudes from $B, C$ to $AC, AB$ respectively. Let $D$ be the midpoint of $BC$. Let $\omega_1$ be the circle with diameter $AD$. Let $Q\neq A$ be the intersection of $(ABC)$ and $\omega$. Let $H$ be the orthocenter of $ABC$. Let $K$ be the intersection of $AQ$ and $BC$. Let $l_1,l_2$ be the lines through $Q$ tangent to $\omega,(AYZ)$ respectively. Let $I$ be the intersection of $l_1$ and $KH$. Let $P$ be the intersection of $l_2$ and $YZ$. Let $l$ be the line through $I$ parallel to $HD$ and let $O'$ be the reflection of $O$ across $l$. Prove that $O'P$ is tangent to $(KPQ)$.
2 replies
vincentwant
Today at 3:54 PM
MathLuis
3 hours ago
J
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Functional Geometry
GreekIdiot   2
N 3 hours ago by Double07
Source: BMO 2024 SL G7
Let $f: \pi \to \mathbb R$ be a function from the Euclidean plane to the real numbers such that $f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$ for any acute triangle $\Delta ABC$ with circumcenter $O$, centroid $G$ and orthocenter $H$. Prove that $f$ is constant.
2 replies
GreekIdiot
Apr 27, 2025
Double07
3 hours ago
J
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Right-angled triangle if circumcentre is on circle
liberator   78
N 3 hours ago by bin_sherlo
Source: IMO 2013 Problem 3
Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcentre of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled.

Proposed by Alexander A. Polyansky, Russia
78 replies
liberator
Jan 4, 2016
bin_sherlo
3 hours ago
J
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J
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Sum of distances in tetrahedron
DPopov   6
N Jun 12, 2022 by Davsch
Source: IMO 1966, Day 1, Problem 3
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.
6 replies
DPopov
Oct 14, 2005
Davsch
Jun 12, 2022
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Sum of distances in tetrahedron
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Reveal topic tags
geometry
3D geometry
tetrahedron
sphere
IMO
IMO 1966
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G H BBookmark kLocked kLocked NReply
Source: IMO 1966, Day 1, Problem 3
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DPopov
1398 posts
DPopov
#1 h  Oct 14, 2005, 7:22 PM • 3 Y
Y by Adventure10, ImSh95, Amir Hossein
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space.
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matematikros
4 posts
matematikros
#2 h  Apr 24, 2009, 2:27 AM • 3 Y
Y by ImSh95, Adventure10, Mango247
Click to reveal hidden text
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tuan119
1157 posts
tuan119
#3 h  May 18, 2011, 4:43 PM • 3 Y
Y by ImSh95, Adventure10, Mango247
matematikros wrote:
Click to reveal hidden text
//cdn.artofproblemsolving.com/images/a7c7b5fbf7e99dd498ea3556cd95204b738663c9.jpg
This explanation does not seem right. because, subject to all of this tetrahedron $ABCD$ is regular tetrahedron.
With your solution, just a special case of the tetrahedron $ABCD$, not true for the general:$MA+MB+MC+MD \ge 4R$.
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sunjae62700
21 posts
sunjae62700
#4 h  Jan 18, 2016, 4:05 AM • 2 Y
Y by ImSh95, Adventure10
matematikros wrote:
Click to reveal hidden text

Why does this hold??
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mohamed10osama
4 posts
mohamed10osama
#5 h  Nov 10, 2020, 12:26 PM • 1 Y
Y by ImSh95
because quadratic mean greater than or equal to algebric mean
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mohamed10osama
4 posts
mohamed10osama
#6 h  Nov 10, 2020, 12:26 PM • 1 Y
Y by ImSh95
But I wonder why he make equivalent of AM with QM
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Davsch
381 posts
Davsch
#7 h  Jun 12, 2022, 10:31 AM • 1 Y
Y by ImSh95
I'm pretty sure the above solution is incorrect, since AM<=QM, not the other way around.


Let $ABCD$ be the tetrahedron and $P$ be any point in space. We want to find the optimal point $P$. We let $P$ vary on the plane passing through it parallel to $ABC$. Then obviously, the distance $PD$ is minimal when $P$ lies on line $DO$, where $O$ is the centroid of the tetrahedron. We claim that $PA+PB+PC$ is then minimal as well, which will finish the problem (since then $P$ analogously lies on $BO$, so $P=O$).

To show this, we now fix $PC$, so that $P$ varies on a fixed circle, the center of which is equidistant from $A,B$. Now consider the set of points $M$ for which $AM+BM$ is fixed. This is an ellipsoid and thus, a convex set. We now consider growing ellipsoids and the minimum $AP+BP$ for $P$ on the circle is obviously achieved for the point where the ellipsoid is first tangent to the circle. Assume that this point does not lie symmetrically with the plane $\mathcal P_{CDO}$. Then we could reflect the minimum across this plane. By symmetry, this point has the same sum of distances. By convexity, their midpoint lies inside the ellipsoid and thus, it has a smaller sum of distances. However, the point where $\mathcal P_{CDO}$ intersects the circle (the intersection point closer to $A,B$), is obviously closer to $A,B$ than this midpoint. Hence, we conclude $O$ is indeed the desired point.
This post has been edited 2 times. Last edited by Davsch, Jun 12, 2022, 10:42 AM
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