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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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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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
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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RMM 2019 Problem 2
math90   79
N a minute ago by lpieleanu
Source: RMM 2019
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.

Jakob Jurij Snoj, Slovenia
79 replies
math90
Feb 23, 2019
lpieleanu
a minute ago
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Good divisors and special numbers.
Nuran2010   2
N 7 minutes ago by BR1F1SZ
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
$N$ is a positive integer. Call all positive divisors of $N$ which are different from $1$ and $N$ beautiful divisors.We call $N$ a special number when it has at least $2$ beautiful divisors and difference of any $2$ beautiful divisors divides $N$ as well. Find all special numbers.
2 replies
Nuran2010
Today at 4:52 PM
BR1F1SZ
7 minutes ago
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Inequality with 3 variables and a special condition
Nuran2010   2
N 27 minutes ago by Assassino9931
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
For positive real numbers $a,b,c$ we have $3abc \geq ab+bc+ca$.
Prove that:

$\frac{1}{a^3+b^3+c}+\frac{1}{b^3+c^3+a}+\frac{1}{c^3+a^3+b} \leq \frac{3}{a+b+c}$.

Determine the equality case.
2 replies
Nuran2010
Today at 5:06 PM
Assassino9931
27 minutes ago
J
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weird Condition
B1t   5
N an hour ago by Sadigly
Source: Mongolian TST 2025 P4
In triangle \(ABC\), where \(AC < AB\), the internal angle bisectors of angles \(\angle A\), \(\angle B\), and \(\angle C\) meet the sides \(BC\), \(AC\), and \(AB\) at points \(D\), \(E\), and \(F\), respectively. Let \( I \) be the incenter of triangle \( AEF \), and let \( G \) be the foot of the perpendicular from \( I \) to line \( BC \). Prove that if the quadrilateral \( DGEF \) is cyclic, then the center of its circumcircle lies on segment \( AD \).
5 replies
B1t
Apr 27, 2025
Sadigly
an hour ago
J
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Circumcircle of one triangle passes from another's circumcenter.
Nuran2010   1
N an hour ago by Assassino9931
Source: Azerbaijan Al-Khwarizmi IJMO TST 2024
In a parallelogram $ABCD$,$\angle A<90^\circ$ and $AB<BC$. Interior angle bisector of $\angle BAD$ intersects $BC$ at $M$, and $DC$ at $N$.Prove that circumcircle of $BCD$ passes from circumcenter of $CMN$.
1 reply
Nuran2010
Today at 4:57 PM
Assassino9931
an hour ago
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a_i/i sequence
pad   19
N an hour ago by ihatemath123
Source: TSTST 2021/2
Let $a_1<a_2<a_3<a_4<\cdots$ be an infinite sequence of real numbers in the interval $(0,1)$. Show that there exists a number that occurs exactly once in the sequence
\[ \frac{a_1}{1},\frac{a_2}{2},\frac{a_3}{3},\frac{a_4}{4},\ldots.\]
Merlijn Staps
19 replies
pad
Nov 8, 2021
ihatemath123
an hour ago
J
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Find points with sames integer distances as given
nAalniaOMliO   2
N an hour ago by nAalniaOMliO
Source: Belarus TST 2024
Points $A_1, \ldots A_n$ with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points $B_1, \ldots ,B_n$ with integer coordinates such that $A_iA_j=B_iB_j$ for every pair $1 \leq i \leq j \leq n$
N. Sheshko, D. Zmiaikou
2 replies
nAalniaOMliO
Jul 17, 2024
nAalniaOMliO
an hour ago
J
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Geometry tangent circles
Stefan4024   68
N an hour ago by zuat.e
Source: EGMO 2016 Day 2 Problem 4
Two circles $\omega_1$ and $\omega_2$, of equal radius intersect at different points $X_1$ and $X_2$. Consider a circle $\omega$ externally tangent to $\omega_1$ at $T_1$ and internally tangent to $\omega_2$ at point $T_2$. Prove that lines $X_1T_1$ and $X_2T_2$ intersect at a point lying on $\omega$.
68 replies
Stefan4024
Apr 13, 2016
zuat.e
an hour ago
J
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My Unsolved Problem
MinhDucDangCHL2000   2
N 2 hours ago by hukilau17
Source: 2024 HSGS Olympiad
Let triangle $ABC$ be inscribed in the circle $(O)$. A line through point $O$ intersects $AC$ and $AB$ at points $E$ and $F$, respectively. Let $P$ be the reflection of $E$ across the midpoint of $AC$, and $Q$ be the reflection of $F$ across the midpoint of $AB$. Prove that:
a) the reflection of the orthocenter $H$ of triangle $ABC$ across line $PQ$ lies on the circle $(O)$.
b) the orthocenters of triangles $AEF$ and $HPQ$ coincide.

Im looking for a solution used complex bashing :(
2 replies
MinhDucDangCHL2000
Today at 4:53 PM
hukilau17
2 hours ago
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Easy Combinatorics
MuradSafarli   2
N 2 hours ago by Sadigly
A student firstly wrote $x=3$ on the board. For each procces, the stutent deletes the number x and replaces it with either $(2x+4)$ or $(3x+8)$ or $(x^2+5x)$. Is this possible to make the number $(20^{25}+2024)$ on the board?
2 replies
MuradSafarli
5 hours ago
Sadigly
2 hours ago
J
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4 variables with quadrilateral sides 2
mihaig   0
3 hours ago
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
0 replies
mihaig
3 hours ago
0 replies
J
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Number theory
MuradSafarli   1
N 3 hours ago by Sadigly
Prove that for any natural number \( n \) :

\[ 1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3). \]
1 reply
MuradSafarli
4 hours ago
Sadigly
3 hours ago
J
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D1025 : Can you do that?
Dattier   0
3 hours ago
Source: les dattes à Dattier
Let $x_{n+1}=x_n^3$ and $x_0=3$.

Can you calculate $\sum\limits_{i=1}^{2^{2025}} x_i \mod 10^{30}$?
0 replies
Dattier
3 hours ago
0 replies
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Perpendicularity
April   32
N 3 hours ago by zuat.e
Source: CGMO 2007 P5
Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.
32 replies
April
Dec 28, 2008
zuat.e
3 hours ago
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Playing cards 1
prof.   1
N Apr 17, 2025 by prof.
In how many ways can a deck of 52 cards be divided among 13 players, each with 4 cards, so that one player has all 4 suits and the others have one suit?
1 reply
prof.
Apr 15, 2025
prof.
Apr 17, 2025
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Playing cards 1
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Reveal topic tags
combinatorics
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prof.
444 posts
prof.
#1 h  Apr 15, 2025, 5:59 AM
Y by
In how many ways can a deck of 52 cards be divided among 13 players, each with 4 cards, so that one player has all 4 suits and the others have one suit?
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prof.
444 posts
prof.
#2 h  Apr 17, 2025, 12:40 PM
Y by
Reminder :blush:
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