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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.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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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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Acute Triangle
Jackson0423   0
6 minutes ago
Source: Putnam

For each \( i = 1, 2, 3, \ldots, 12 \), let \( d_i \) be a real number such that \( 1 \leq d_i \leq 12 \).
Prove that no matter how the numbers \( d_1, d_2, \ldots, d_{12} \) are chosen,
there exist indices \( 1 \leq a < b < c \leq 12 \) such that \( d_a, d_b, d_c \) can be the side lengths of an acute triangle.
0 replies
Jackson0423
6 minutes ago
0 replies
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Real numbers
Jackson0423   1
N 15 minutes ago by Primeniyazidayi

Among any \( n \) distinct real numbers, there exist two numbers \( a \) and \( b \) such that
\[ a^2 + b^2 < \sqrt{3}(1 + a^2b^2). \]Find the smallest possible value of \( n \).
1 reply
Jackson0423
39 minutes ago
Primeniyazidayi
15 minutes ago
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IMO Shortlist 2011, Algebra 4
orl   18
N 16 minutes ago by shanelin-sigma
Source: IMO Shortlist 2011, Algebra 4
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.

Proposed by Bojan Bašić, Serbia
18 replies
orl
Jul 11, 2012
shanelin-sigma
16 minutes ago
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My unsolved problem
ZeltaQN2008   0
23 minutes ago
Source: Belarus 2017
Find all funcition $f:(0,\infty)\rightarrow (0,\infty)$ such that for all any $x,y\in (0,\infty)$ :
$f(x+f(xy))=xf(1+f(y))$
0 replies
ZeltaQN2008
23 minutes ago
0 replies
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Minimizing Triangle Area
v_Power   1
N 26 minutes ago by lele0305
Hi,

I am trying to solve this question:
A square piece of toast ABCD of side length 1 and centre O is cut in half to form two equal pieces ABC and CDA. If the triangle ABC has to be cut into two parts of equal area, one would usually cut along the line of symmetry BO. However, there are other ways of doing this. Find, with justification, the length and location of the shortest straight cut which divides the triangle ABC into two parts of equal area.
I have worked out that the length is sqrt(sqrt(2)-1) using calculus, but I was wondering if there was a way to solve it without using calculus?
1 reply
v_Power
an hour ago
lele0305
26 minutes ago
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Prove XBY equal to angle C
nataliaonline75   1
N an hour ago by cj13609517288
Let $M$ be the midpoint of $BC$ on triangle $ABC$. Point $X$ lies on segment $AC$ such that $AX=BX$ and $Y$ on line $AM$ such that $XY//AB$. Prove that $\angle XBY = \angle ACB$.
1 reply
nataliaonline75
an hour ago
cj13609517288
an hour ago
J
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Inequality related to geometry
ducthien   0
an hour ago
Let \( ABCD \) be a convex quadrilateral. The diagonals \( AC \) and \( BD \) intersect at \( P \), with \( \angle APD = 60^\circ \). Let \( E, F, G, \) and \( H \) be the midpoints of sides \( AB, BC, CD \), and \( DA \), respectively. Find the greatest positive real number \( k \) such that
\[ EG + 3FH \geq k \cdot d + (1 - k) \cdot s, \]where \( s \) is the semiperimeter of \( ABCD \) and \( d \) is the sum of the lengths of its diagonals (i.e., \( d = AC + BD \)). Determine when equality holds.

Im trying to slove this question
0 replies
ducthien
an hour ago
0 replies
J
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trig relation with two equal circles, each passing through center of other
parmenides51   4
N an hour ago by Captainscrubz
Source: Sharygin 2010 Final 10.2
Each of two equal circles $\omega_1$ and $\omega_2$ passes through the center of the other one. Triangle $ABC$ is inscribed into $\omega_1$, and lines $AC, BC$ touch $\omega_2$ . Prove that $cosA + cosB = 1$.
4 replies
parmenides51
Nov 25, 2018
Captainscrubz
an hour ago
J
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angle chasing, tangents at circumcircle of a right triangle
parmenides51   1
N 2 hours ago by CovertQED
Source: China Northern MO 2015 10.2 CNMO
It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.
1 reply
parmenides51
Oct 28, 2022
CovertQED
2 hours ago
J
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Showing Tangency
Itoz   0
2 hours ago
Source: Own
The circumcenter of $\triangle ABC$ is $O$. Line $AO$ meets line $BC$ at point $D$, and there is a point $E$ on $\odot(ABC)$ such that $AE \perp BC$. Line $DE$ intersects $\odot(ABC)$ at point $F$. The perpendicular bisector of line segment $BC$ intersects line $AB$ at point $K$, and line $AB$ intersects $\odot(CFK)$ at point $L$.

Prove that $\odot(AFL)$ is tangent to $\odot (OBC)$.
0 replies
Itoz
2 hours ago
0 replies
J
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Perpendicularity
April   33
N 3 hours ago by AshAuktober
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 $DE \perp EF$.
33 replies
April
Dec 28, 2008
AshAuktober
3 hours ago
J
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<HKG = 90^o if AG = GH
parmenides51   15
N 3 hours ago by complex2math
Source: 2020 Balkan MO shortlist G2
Let $G, H$ be the centroid and orthocentre of $\vartriangle ABC$ which has an obtuse angle at $\angle B$. Let $\omega$ be the circle with diameter $AG$. $\omega$ intersects $\odot(ABC)$ again at $L \ne A$. The tangent to $\omega$ at $L$ intersects $\odot(ABC)$ at $K \ne L$. Given that $AG = GH$, prove $\angle HKG = 90^o$
.
Sam Bealing, United Kingdom
15 replies
parmenides51
Sep 14, 2021
complex2math
3 hours ago
J
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Geometry with orthocenter config
thdnder   5
N 3 hours ago by thdnder
Source: Own
Let $ABC$ be a triangle, and let $AD, BE, CF$ be its altitudes. Let $H$ be its orthocenter, and let $O_B$ and $O_C$ be the circumcenters of triangles $AHC$ and $AHB$. Let $G$ be the second intersection of the circumcircles of triangles $FDO_B$ and $EDO_C$. Prove that the lines $DG$, $EF$, and $A$-median of $\triangle ABC$ are concurrent.
5 replies
thdnder
Apr 29, 2025
thdnder
3 hours ago
J
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A coincidence about triangles with common incenter
flower417477   7
N 3 hours ago by flower417477
$\triangle ABC,\triangle ADE$ have the same incenter $I$.Prove that $BCDE$ is concyclic iff $BC,DE,AI$ is concurrent
7 replies
flower417477
Apr 30, 2025
flower417477
3 hours ago
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Funny function that there isn't exist
ItzsleepyXD   5
N Apr 26, 2025 by Hamzaachak
Source: Own, Modified from old problem
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
5 replies
ItzsleepyXD
Apr 10, 2025
Hamzaachak
Apr 26, 2025
J
Funny function that there isn't exist
G H J
Reveal topic tags
function
number theory
functional equation
Divisibility
Euler s Phi Function
L
G H BBookmark kLocked kLocked NReply
Source: Own, Modified from old problem
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ItzsleepyXD
130 posts
ItzsleepyXD
#1 h  Apr 10, 2025, 1:50 AM
Y by
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$,
$$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$
Z K Y
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ItzsleepyXD
130 posts
ItzsleepyXD
#2 h  Apr 25, 2025, 4:06 AM
Y by
Bump....
Z K Y
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EvansGressfield
3 posts
EvansGressfield
#3 h  Apr 26, 2025, 4:17 PM
Y by
any idea?
Z K Y
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Rayanelba
14 posts
Rayanelba
#4 h  Apr 26, 2025, 6:39 PM
Y by
This problem is very hard pls share some hints
Z K Y
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Hamzaachak
61 posts
Hamzaachak
#5 h  Apr 26, 2025, 8:48 PM
Y by
No such function exist
Z K Y
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Hamzaachak
61 posts
Hamzaachak
#6 h  Apr 26, 2025, 9:26 PM • 1 Y
Y by ItzsleepyXD
Hamzaachak wrote:
No such function exist

Small hint : try to prove f(2)=0
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