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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.

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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
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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How many triangles
Ecrin_eren   4
N an hour ago by Evanlovemath


"Inside a triangle, 2025 points are placed, and each point is connected to the vertices of the smallest triangle that contains it. In the final state, how many small triangles are formed?"


4 replies
Ecrin_eren
May 2, 2025
Evanlovemath
an hour ago
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Triangle on a tetrahedron
vanstraelen   2
N 3 hours ago by ReticulatedPython

Given a regular tetrahedron $(A,BCD)$ with edges $l$.
Construct at the apex $A$ three perpendiculars to the three lateral faces.
Take a point on each perpendicular at a distance $l$ from the apex such that these three points lie above the apex.
Calculate the lenghts of the sides of the triangle.
2 replies
vanstraelen
Today at 2:43 PM
ReticulatedPython
3 hours ago
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Problem 4 of Finals
GeorgeRP   2
N 3 hours ago by Assassino9931
Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade
The diagonals \( AD \), \( BE \), and \( CF \) of a hexagon \( ABCDEF \) inscribed in a circle \( k \) intersect at a point \( P \), and the acute angle between any two of them is \( 60^\circ \). Let \( r_{AB} \) be the radius of the circle tangent to segments \( PA \) and \( PB \) and internally tangent to \( k \); the radii \( r_{BC} \), \( r_{CD} \), \( r_{DE} \), \( r_{EF} \), and \( r_{FA} \) are defined similarly. Prove that
\[ r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}. \]
2 replies
GeorgeRP
Sep 10, 2024
Assassino9931
3 hours ago
J
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Interesting functional equation with geometry
User21837561   3
N 4 hours ago by Double07
Source: BMOSL 2024 G7
For an acute triangle $ABC$, let $O$ be the circumcentre, $H$ be the orthocentre, and $G$ be the centroid.
Let $f:\pi\rightarrow\mathbb R$ satisfy the following condition:
$f(A)+f(B)+f(C)=f(O)+f(G)+f(H)$
Prove that $f$ is constant.
3 replies
User21837561
Today at 8:14 AM
Double07
4 hours ago
J
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shadow of a cylinder, shadow of a cone
vanstraelen   2
N 4 hours ago by vanstraelen

a) Given is a right cylinder of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the botom base?

b) Given is a right cone of height $2R$ and radius $R$.
The sun shines on this solid at an angle of $45^{\circ}$.
What is the area of the shadow that this solid casts on the plane of the base?
2 replies
vanstraelen
Today at 3:08 PM
vanstraelen
4 hours ago
J
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greatest volume
hzbrl   1
N 4 hours ago by hzbrl
Source: purple comet
A large sphere with radius 7 contains three smaller balls each with radius 3 . The three balls are each externally tangent to the other two balls and internally tangent to the large sphere. There are four right circular cones that can be inscribed in the large sphere in such a way that the bases of the cones are tangent to all three balls. Of these four cones, the one with the greatest volume has volume $n \pi$. Find $n$.
1 reply
hzbrl
Yesterday at 9:56 AM
hzbrl
4 hours ago
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(n+1)2^n, (n+3)2^{n+2} not perfect squares for the same n
parmenides51   3
N 4 hours ago by AylyGayypow009
Source: Greece JBMO TST 2015 p3
Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.
3 replies
parmenides51
Apr 29, 2019
AylyGayypow009
4 hours ago
J
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IMO 2010 Problem 3
canada   59
N 4 hours ago by pi271828
Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$

Proposed by Gabriel Carroll, USA
59 replies
canada
Jul 7, 2010
pi271828
4 hours ago
J
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Equilateral triangle formed by circle and Fermat point
Mimii08   2
N 4 hours ago by Mimii08
Source: Heard from a friend
Hi! I found this interesting geometry problem and I would really appreciate help with the proof.

Let ABC be an acute triangle, and let T be the Fermat (Torricelli) point of triangle ABC. Let A1, B1, and C1 be the feet of the perpendiculars from T to the sides BC, AC, and AB, respectively. Let ω be the circle passing through points A1, B1, and C1. Let A2, B2, and C2 be the second points where ω intersects the sides BC, AC, and AB, respectively (different from A1, B1, C1).

Prove that triangle A2B2C2 is equilateral.

2 replies
Mimii08
Yesterday at 10:36 PM
Mimii08
4 hours ago
J
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two circumcenters and one orthocenter, vertices of parallelogram
parmenides51   4
N 4 hours ago by AylyGayypow009
Source: Greece JBMO TST 2015 p2
Let $ABC$ be an acute triangle inscribed in a circle of center $O$. If the altitudes $BD,CE$ intersect at $H$ and the circumcenter of $\triangle BHC$ is $O_1$, prove that $AHO_1O$ is a parallelogram.
4 replies
parmenides51
Apr 29, 2019
AylyGayypow009
4 hours ago
J
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m^4+3^m is a perfect square number
Havu   5
N 4 hours ago by MR.1
Find a positive integer m such that $m^4+3^m$ is a perfect square number.
5 replies
Havu
6 hours ago
MR.1
4 hours ago
J
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Determine all the 'good' numbers
April   4
N 5 hours ago by DottedCaculator
Source: CGMO 2004 P1
We say a positive integer $ n$ is good if there exists a permutation $ a_1, a_2, \ldots, a_n$ of $ 1, 2, \ldots, n$ such that $ k + a_k$ is perfect square for all $ 1\le k\le n$. Determine all the good numbers in the set $ \{11, 13, 15, 17, 19\}$.
4 replies
April
Dec 27, 2008
DottedCaculator
5 hours ago
J
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Inequalities
sqing   6
N 5 hours ago by hi2024IMOp14
Let $ a,b>0, a^2+ab+b^2 \geq 6 $. Prove that
$$a^4+ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$a^4+ab+b^4 \leq 10$$Let $ a,b>0, a^2+ab+b^2 \geq \frac{15}{2} $. Prove that
$$ a^4-ab+b^4\geq 10$$Let $ a,b>0, a^2+ab+b^2 \leq \sqrt{10} $. Prove that
$$-\frac{1}{8}\leq a^4-ab+b^4\leq 10$$
6 replies
sqing
Yesterday at 2:42 PM
hi2024IMOp14
5 hours ago
J
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Classical factorial number theory
Orestis_Lignos   21
N 5 hours ago by MIC38
Source: JBMO 2023 Problem 1
Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$.

Nikola Velov, North Macedonia
21 replies
Orestis_Lignos
Jun 26, 2023
MIC38
5 hours ago
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J
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BMT 2018 Algebra Round Problem 7
IsabeltheCat   5
N Apr 22, 2025 by P162008
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
5 replies
IsabeltheCat
Dec 3, 2018
P162008
Apr 22, 2025
J
BMT 2018 Algebra Round Problem 7
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Reveal topic tags
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IsabeltheCat
4242 posts
IsabeltheCat
#1 h  Dec 3, 2018, 3:36 AM • 2 Y
Y by Adventure10, Mango247
Let $$h_n := \sum_{k=0}^n \binom{n}{k} \frac{2^{k+1}}{(k+1)}.$$Find $$\sum_{n=0}^\infty \frac{h_n}{n!}.$$
This post has been edited 1 time. Last edited by IsabeltheCat, Dec 3, 2018, 11:11 PM
Reason: wrong dummy variable was used previously
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rzlng
99 posts
rzlng
#2 h  Dec 3, 2018, 4:51 AM • 3 Y
Y by Srofller, Adventure10, Mango247
sketch
@below, thanks. fixed
This post has been edited 1 time. Last edited by rzlng, Dec 3, 2018, 5:11 AM
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TomCalc
1635 posts
TomCalc
#3 h  Dec 3, 2018, 5:05 AM • 1 Y
Y by Adventure10
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.
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NikoIsLife
9657 posts
NikoIsLife
#4 h  Dec 3, 2018, 10:59 AM • 3 Y
Y by Adventure10, Mango247, soryn
Solution
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IsabeltheCat
4242 posts
IsabeltheCat
#5 h  Dec 3, 2018, 11:11 PM • 2 Y
Y by Adventure10, Mango247
TomCalc wrote:
IsabeltheCat wrote:
Find $$\sum_{k=0}^\infty \frac{h_n}{n!}.$$
The second sum has a wrong dummy variable and @above the answer is correct.

Thanks for pointing that out. I fixed the error in the original post.
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P162008
187 posts
P162008
#6 h  Apr 22, 2025, 6:56 AM
Y by
Storage
This post has been edited 3 times. Last edited by P162008, Apr 28, 2025, 9:40 PM
Reason: Typo
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