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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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n = d2^2 + d3^3
codyj   4
N 33 minutes ago by Namisgood
Source: OMM 2008 1
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$.
4 replies
codyj
Jul 19, 2014
Namisgood
33 minutes ago
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Another Number Theory!
matinyousefi   7
N 43 minutes ago by MR.1
Source: Iran MO 2024 second round P6
Find all natural numbers $x,y>1$and primes $p$ that satisfy $$\frac{x^2-1}{y^2-1}=(p+1)^2. $$
7 replies
1 viewing
matinyousefi
Apr 19, 2024
MR.1
43 minutes ago
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IMO Shortlist 2014 A2
hajimbrak   40
N an hour ago by ezpotd
Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\]

Proposed by Denmark
40 replies
hajimbrak
Jul 11, 2015
ezpotd
an hour ago
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sequence positive
malinger   38
N an hour ago by ezpotd
Source: ISL 2006, A2, VAIMO 2007, P4, Poland 2007
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$.

Proposed by Mariusz Skalba, Poland
38 replies
malinger
Apr 22, 2007
ezpotd
an hour ago
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3 numbers have their fractional parts lying in the interval
orl   13
N 2 hours ago by ezpotd
Source: IMO Shortlist 2000, A2
Let $ a, b, c$ be positive integers satisfying the conditions $ b > 2a$ and $ c > 2b.$ Show that there exists a real number $ \lambda$ with the property that all the three numbers $ \lambda a, \lambda b, \lambda c$ have their fractional parts lying in the interval $ \left(\frac {1}{3}, \frac {2}{3} \right].$
13 replies
orl
Aug 10, 2008
ezpotd
2 hours ago
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IMO 2016 Problem 2
shinichiman   65
N 2 hours ago by Mathgloggers
Source: IMO 2016 Problem 2
Find all integers $n$ for which each cell of $n \times n$ table can be filled with one of the letters $I,M$ and $O$ in such a way that:
[LIST]
[*] in each row and each column, one third of the entries are $I$, one third are $M$ and one third are $O$; and [/*]
[*]in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are $I$, one third are $M$ and one third are $O$.[/*]
[/LIST]
Note. The rows and columns of an $n \times n$ table are each labelled $1$ to $n$ in a natural order. Thus each cell corresponds to a pair of positive integer $(i,j)$ with $1 \le i,j \le n$. For $n>1$, the table has $4n-2$ diagonals of two types. A diagonal of first type consists all cells $(i,j)$ for which $i+j$ is a constant, and the diagonal of this second type consists all cells $(i,j)$ for which $i-j$ is constant.
65 replies
shinichiman
Jul 11, 2016
Mathgloggers
2 hours ago
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Proving the line is indeed a radical axis
azzam2912   1
N 2 hours ago by JARP091
Given an acute triangle ABC with altitudes AD, BE, and CF intersecting at point H. Let O be the center of the circumcircle of triangle ABC. The Tangents to the circumcircle of triangle ABC from points B and C intersect at point T. Let K and L be reflections of point O on lines AB and AC respectively. The circumcircles of triangle DFK and DEL intersect a second time at point P. Prove that points P, D, and T are collinear.
1 reply
azzam2912
2 hours ago
JARP091
2 hours ago
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A point on BC
jayme   0
2 hours ago
Source: Own ?
Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.

Prove : B, C and T are collinear.

Sincerely
Jean-Louis
0 replies
jayme
2 hours ago
0 replies
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Balkan Mathematical Olympiad
ABCD1728   0
3 hours ago
Can anyone provide the PDF version of the book "Balkan Mathematical Olympiads" by Mircea Becheanu and Bogdan Enescu (published by XYZ press in 2014), thanks!
0 replies
ABCD1728
3 hours ago
0 replies
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A sharp one with 3 var
mihaig   4
N 3 hours ago by arqady
Source: Own
Let $a,b,c\geq0$ satisfying
$$\left(a+b+c-2\right)^2+8\leq3\left(ab+bc+ca\right).$$Prove
$$ab+bc+ca+abc\geq4.$$
4 replies
mihaig
May 13, 2025
arqady
3 hours ago
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Find all p(x) such that p(p) is a power of 2
truongphatt2668   5
N 4 hours ago by tom-nowy
Source: ???
Find all polynomial $P(x) \in \mathbb{R}[x]$ such that:
$$P(p_i) = 2^{a_i}$$with $p_i$ is an $i$ th prime and $a_i$ is an arbitrary positive integer.
5 replies
truongphatt2668
Thursday at 1:05 PM
tom-nowy
4 hours ago
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Interesting problem from a friend
v4913   10
N 4 hours ago by OronSH
Source: I'm not sure...
Let the incircle $(I)$ of $\triangle{ABC}$ touch $BC$ at $D$, $ID \cap (I) = K$, let $\ell$ denote the line tangent to $(I)$ through $K$. Define $E, F \in \ell$ such that $\angle{EIF} = 90^{\circ}, EI, FI \cap (AEF) = E', F'$. Prove that the circumcenter $O$ of $\triangle{ABC}$ lies on $E'F'$.
10 replies
+1 w
v4913
Nov 25, 2023
OronSH
4 hours ago
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IMO ShortList 2002, algebra problem 3
orl   25
N 4 hours ago by Mathandski
Source: IMO ShortList 2002, algebra problem 3
Let $P$ be a cubic polynomial given by $P(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are integers and $a\ne0$. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
25 replies
orl
Sep 28, 2004
Mathandski
4 hours ago
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Inequality on APMO P5
Jalil_Huseynov   41
N 4 hours ago by Mathandski
Source: APMO 2022 P5
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.
41 replies
Jalil_Huseynov
May 17, 2022
Mathandski
4 hours ago
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maximal cube in tetrahedron, one vertex on feet of altitude
jasperE3   0
Jun 15, 2021
Source: Bulgaria 1986 P3
A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
0 replies
jasperE3
Jun 15, 2021
0 replies
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maximal cube in tetrahedron, one vertex on feet of altitude
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Source: Bulgaria 1986 P3
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jasperE3
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jasperE3
#1 h  Jun 15, 2021, 4:56 PM
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A regular tetrahedron of unit edge is given. Find the volume of the maximal cube contained in the tetrahedron, whose one vertex lies in the feet of an altitude of the tetrahedron.
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