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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 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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unfair coin, points winning 2024 TMC AIME Mock #9
parmenides51   5
N 4 minutes ago by Math-lover1
Krithik has an unfair coin with a $\frac13$ chance of landing heads when flipped. Krithik is playing a game where he starts with $1$ point. Every turn, he flips the coin, and if it lands heads, he gains $1$ point, and if it lands tails, he loses $1$ point. However, after the turn, if he has a negative number of points, his point counter resets to $1$. Krithik wins when he earns $8$ points. Find the expected number of turns until Krithik wins.
5 replies
parmenides51
Apr 26, 2025
Math-lover1
4 minutes ago
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Can you construct the incenter of a triangle ABC?
PennyLane_31   3
N 44 minutes ago by cj13609517288
Source: 2023 Girls in Mathematics Tournament- Level B, Problem 4
Given points $P$ and $Q$, Jaqueline has a ruler that allows tracing the line $PQ$. Jaqueline also has a special object that allows the construction of a circle of diameter $PQ$. Also, always when two circles (or a circle and a line, or two lines) intersect, she can mark the points of the intersection with a pencil and trace more lines and circles using these dispositives by the points marked. Initially, she has an acute scalene triangle $ABC$. Show that Jaqueline can construct the incenter of $ABC$.
3 replies
PennyLane_31
Oct 29, 2023
cj13609517288
44 minutes ago
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BrUMO 2025 Team Round Problem 15
lpieleanu   1
N an hour ago by vanstraelen
Let $\triangle{ABC}$ be an isosceles triangle with $AB=AC.$ Let $D$ be a point on the circumcircle of $\triangle{ABC}$ on minor arc $AB.$ Let $\overline{AD}$ intersect the extension of $\overline{BC}$ at $E.$ Let $F$ be the midpoint of segment $AC,$ and let $G$ be the intersection of $\overline{EF}$ and $\overline{AB}.$ Let the extension of $\overline{DG}$ intersect $\overline{AC}$ and the circumcircle of $\triangle{ABC}$ at $H$ and $I,$ respectively. Given that $DG=3, GH=5,$ and $HI=1,$ compute the length of $\overline{AE}.$
1 reply
lpieleanu
Apr 27, 2025
vanstraelen
an hour ago
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Do not try to bash on beautiful geometry
ItzsleepyXD   3
N an hour ago by FarrukhBurzu
Source: Own , Mock Thailand Mathematic Olympiad P9
Let $ABC$be triangle with point $D,E$ and $F$ on $BC,AB,CA$
such that $BE=CF$ and $E,F$ are on the same side of $BC$
Let $M$ be midpoint of segment $BC$ and $N$ be midpoint of segment $EF$
Let $G$ be intersection of $BF$ with $CE$ and $\dfrac{BD}{DC}=\dfrac{AC}{AB}$
Prove that $MN\parallel DG$
3 replies
ItzsleepyXD
Today at 9:30 AM
FarrukhBurzu
an hour ago
J
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Cool functional equation
Rayanelba   2
N an hour ago by ATM_
Source: Own
Find all functions $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ that verify the following equation for all $x,y\in \mathbb{Z}_{>0}$:
$max(f^{f(y)}(x),f^{f(y)}(y))|min(x,y)$
2 replies
Rayanelba
an hour ago
ATM_
an hour ago
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1 line solution to Inequality
ItzsleepyXD   2
N an hour ago by Vivaandax
Source: Own , Mock Thailand Mathematic Olympiad P8
Let $x_1,x_2,\dots,x_n$ be positive real integer such that $x_1^2+x_2^2+\cdots+x_n^2=2$ Prove that
$$\sum_{i=1}^{n}\frac{1}{x_i^3(x_{i-1}+x_{i+1})}\geqslant \left(\sum_{i=1}^{n}\frac{x_i}{x_{i-1}+x_{i+1}}\right)^3$$such that $x_{n+1}=x_1$ and $x_0=x_n$
2 replies
ItzsleepyXD
Today at 9:27 AM
Vivaandax
an hour ago
J
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primes,exponentials,factorials
skellyrah   2
N an hour ago by CM1910
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
2 replies
skellyrah
2 hours ago
CM1910
an hour ago
J
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trigonometric functions
VivaanKam   9
N 2 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
9 replies
VivaanKam
Yesterday at 8:29 PM
aok
2 hours ago
J
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a nice prob for number theory
Jackson0423   1
N 2 hours ago by alexheinis
Source: number theory
Let \( n \) be a positive integer, and let its positive divisors be
\[ d_1 < d_2 < \cdots < d_k. \]Define \( f(n) \) to be the number of ordered pairs \( (i, j) \) with \( 1 \le i, j \le k \) such that \( \gcd(d_i, d_j) = 1 \).

Find \( f(3431 \times 2999) \).

Also, find a general formula for \( f(n) \) when
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}, \]where the \( p_i \) are distinct primes and the \( e_i \) are positive integers.
1 reply
Jackson0423
5 hours ago
alexheinis
2 hours ago
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C-B=60 <degrees>
Sasha   27
N 2 hours ago by zuat.e
Source: Moldova TST 2005, IMO Shortlist 2004 geometry problem 3
Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$.

Proposed by Hojoo Lee, Korea
27 replies
Sasha
Apr 10, 2005
zuat.e
2 hours ago
J
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4 variables with quadrilateral sides 2
mihaig   1
N 2 hours ago by mihaig
Source: Own
Let $a,b,c,d\geq0$ satisfying
$$\frac1{a+1}+\frac1{b+1}+\frac1{c+1}+\frac1{d+1}=2.$$Prove
$$\left(a+b+c+d-2\right)^2+8\geq3\left(abc+abd+acd+bcd\right).$$
1 reply
mihaig
Yesterday at 8:47 PM
mihaig
2 hours ago
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Good divisors and special numbers.
Nuran2010   3
N 2 hours ago by Nuran2010
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.
3 replies
Nuran2010
Yesterday at 4:52 PM
Nuran2010
2 hours ago
J
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Functionnal equation
Rayanelba   1
N 2 hours ago by Mrcuberoot
Source: Own
Find all functions $f:\mathbb{R}_{>0}\to \mathbb{R}_{>0}$ that verify the following equation for all $x,y\in \mathbb{R}_{>0}$:
$f(x+yf(x))+f(\frac{x}{y})=\frac{x}{y}+f(x+xy)$
1 reply
Rayanelba
5 hours ago
Mrcuberoot
2 hours ago
J
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Number Theory with set and subset and divisibility
SomeonecoolLovesMaths   3
N 4 hours ago by martianrunner
Let $S = \{ 1,2, \cdots, 100 \}$. Let $A$ be a subset of $S$ such that no sum of three distinct elements of $A$ is divisible by $5$. What is the maximum value of $\mid A \mid$.
3 replies
SomeonecoolLovesMaths
Apr 21, 2025
martianrunner
4 hours ago
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J
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geometry incentre config
Tony_stark0094   2
N Apr 2, 2025 by LearnMath_105
In a triangle $\Delta ABC$ $I$ is the incentre and point $F$ is defined such that $F \in AC$ and $F \in \odot BIC$
prove that $AI$ is the perpendicular bisector of $BF$
2 replies
Tony_stark0094
Apr 1, 2025
LearnMath_105
Apr 2, 2025
J
geometry incentre config
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Reveal topic tags
geometry
incenter
perpendicular bisector
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Tony_stark0094
69 posts
Tony_stark0094
#1 h  Apr 1, 2025, 4:09 PM
Y by
In a triangle $\Delta ABC$ $I$ is the incentre and point $F$ is defined such that $F \in AC$ and $F \in \odot BIC$
prove that $AI$ is the perpendicular bisector of $BF$
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Tony_stark0094
69 posts
Tony_stark0094
#2 h  Apr 2, 2025, 7:38 AM
Y by
Note for me
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LearnMath_105
149 posts
LearnMath_105
#3 h  Apr 2, 2025, 4:05 PM
Y by
Note that we want $\triangle ABI \equiv \triangle AFI$ Clearly $\angle BAI = \angle CAI = \angle FAI$ and $AI=AI$. Since BIFC is cyclic we have
\begin{align*} \angle ABI \equiv & \angle CBI \\ & \angle CFI \\ & \angle AFI \end{align*}where the angles are directed.
This post has been edited 1 time. Last edited by LearnMath_105, Apr 2, 2025, 4:05 PM
Reason: edit
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