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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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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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f(n) <= f(a(G)) + f(b(G))
dangerousliri   5
N 11 minutes ago by awesomeming327.
Source: FEOO, Problem 2, Shortlist C2
Given a group $G$ of people, we define $a(G)$ to be the least number of tables needed such that each person from the group sits in one of them with no two friends sitting on the same table, and $b(G)$ to be the least number of tables needed such that each person from the group sits in one of them with no two enemies sitting on the same table.
Consider all functions $f:\mathbb{N} \to \mathbb{N}$ such that for each group $G$ of $n$ people we have
\[ f(n) \leqslant f(a(G)) + f(b(G)) \quad \text{and} \quad f(2) = 1.\]Find all possible values of $f(2020)$.

Note: We assume that for every pair of people in a group, they are either both friends with each other or both enemies with each other. Also, $\mathbb{N}$ is the set of all positive integers.

Proposed by Demetres Christofides, Cyprus
5 replies
dangerousliri
May 30, 2020
awesomeming327.
11 minutes ago
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f(w^2+x^2+y^2+z^2)=f(w^2+x^2)+f(y^2+z^2)
dangerousliri   23
N 24 minutes ago by awesomeming327.
Source: FEOO, Problem 1, Shortlist N1
Find all functions $f:\mathbb{N}_0\rightarrow\mathbb{R}$ such that,
$$f(w^2+x^2+y^2+z^2)=f(w^2+x^2)+f(y^2+z^2)$$for all non-negative integers $w,x,y$ and $z$.

Note: $\mathbb{N}_0$ is the set of all non-negative integers and $\mathbb{R}$ is the set of all real numbers.

Proposed by Dorlir Ahmeti, Kosovo
23 replies
dangerousliri
May 30, 2020
awesomeming327.
24 minutes ago
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Darboux cubic
srirampanchapakesan   0
26 minutes ago
Source: Own
Let P be a point on the Darboux cubic (or the McCay Cubic ) of triangle ABC.

P1P2P3 is the circumcevian or pedal triangle of P wrt ABC.

Prove that P also lie on the Darboux cubic ( or the McCay Cubic) of P1P2P3 .
0 replies
srirampanchapakesan
26 minutes ago
0 replies
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primitive polyominoes
N.T.TUAN   29
N an hour ago by Disjunction
Source: USAMO 2007
An animal with $n$ cells is a connected figure consisting of $n$ equal-sized cells[1].

A dinosaur is an animal with at least $2007$ cells. It is said to be primitive it its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

(1) Animals are also called polyominoes. They can be defined inductively. Two cells are adjacent if they share a complete edge. A single cell is an animal, and given an animal with $n$ cells, one with $n+1$ cells is obtained by adjoining a new cell by making it adjacent to one or more existing cells.
29 replies
N.T.TUAN
Apr 26, 2007
Disjunction
an hour ago
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Only consecutive terms are coprime
socrates   37
N an hour ago by blueprimes
Source: 7th RMM 2015, Problem 1
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?
37 replies
socrates
Feb 28, 2015
blueprimes
an hour ago
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Cyclic roots are not real, they can't hurt you
anantmudgal09   21
N an hour ago by bjump
Source: INMO 2023 P2
Suppose $a_0,\ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1,\ldots, 100\}$:
$$a_{100+k}x^{100}+100a_{99+k}x^{99}+a_{98+k}x^{98}+a_{97+k}x^{97}+\dots+a_{2+k}x^2+a_{1+k}x+a_k,$$where indices are taken modulo $101$, i.e., $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2,\dots, 100\}$. Show that it is impossible that each of these $101$ polynomials has all its roots real.

Proposed by Prithwijit De
21 replies
anantmudgal09
Jan 15, 2023
bjump
an hour ago
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Erasing the difference of two numbers
BR1F1SZ   2
N an hour ago by sami1618
Source: Austria National MO Part 1 Problem 3
Consider the following game for a positive integer $n$. Initially, the numbers $1, 2, \ldots, n$ are written on a board. In each move, two numbers are selected such that their difference is also present on the board. This difference is then erased from the board. (For example, if the numbers $3,6,11$ and $17$ are on the board, then $3$ can be erased as $6 - 3=3$, or $6$ as $17 - 11=6$, or $11$ as $17 - 6=11$.)

For which values of $n$ is it possible to end with only one number remaining on the board?

(Michael Reitmeir)
2 replies
BR1F1SZ
Monday at 9:48 PM
sami1618
an hour ago
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GCD of terms in a sequence
BBNoDollar   0
3 hours ago
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
3 hours ago
0 replies
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Number Theory
fasttrust_12-mn   13
N 3 hours ago by KTYC
Source: Pan African Mathematics Olympiad P1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
13 replies
fasttrust_12-mn
Aug 15, 2024
KTYC
3 hours ago
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GCD of terms in a sequence
BBNoDollar   0
3 hours ago
Determine the general term of the sequence of non-zero natural numbers (a_n)n≥1, with the property that gcd(a_m, a_n, a_p) = gcd(m^2 ,n^2 ,p^2), for any distinct non-zero natural numbers m, n, p.

⁡Note that gcd(a,b,c) denotes the greatest common divisor of the natural numbers a,b,c .
0 replies
BBNoDollar
3 hours ago
0 replies
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Aime type Geo
ehuseyinyigit   3
N 3 hours ago by sami1618
Source: Turkish First Round 2024
In a scalene triangle $ABC$, let $M$ be the midpoint of side $BC$. Let the line perpendicular to $AC$ at point $C$ intersect $AM$ at $N$. If $(BMN)$ is tangent to $AB$ at $B$, find $AB/MA$.
3 replies
ehuseyinyigit
Monday at 9:04 PM
sami1618
3 hours ago
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minimizing sum
gggzul   1
N 4 hours ago by RedFireTruck
Let $x, y, z$ be real numbers such that $x^2+y^2+z^2=1$. Find
$$min\{12x-4y-3z\}.$$
1 reply
gggzul
5 hours ago
RedFireTruck
4 hours ago
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Equilateral Triangle inside Equilateral Triangles.
abhisruta03   2
N 4 hours ago by Reacheddreams
Source: ISI 2021 P6
If a given equilateral triangle $\Delta$ of side length $a$ lies in the union of five equilateral triangles of side length $b$, show that there exist four equilateral triangles of side length $b$ whose union contains $\Delta$.
2 replies
abhisruta03
Jul 18, 2021
Reacheddreams
4 hours ago
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USAMO 1984 Problem 5 - Polynomial of degree 3n
Binomial-theorem   8
N 4 hours ago by Assassino9931
Source: USAMO 1984 Problem 5
$P(x)$ is a polynomial of degree $3n$ such that

\begin{eqnarray*} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{eqnarray*}

Determine $n$.
8 replies
Binomial-theorem
Aug 16, 2011
Assassino9931
4 hours ago
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sequence
dno1467   1
N Apr 27, 2025 by dno1467
Source: Balkan MO Shortlist 2024 N4
Find all sequences $a_n$ of positive integers such that

$a_{n+2}(a_{n+1} - k) = a_n(a_{n+1} + k)$

for all positive integers n.
1 reply
dno1467
Apr 27, 2025
dno1467
Apr 27, 2025
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sequence
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Reveal topic tags
BMO Shortlist
number theory
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Source: Balkan MO Shortlist 2024 N4
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dno1467
5 posts
dno1467
#1 h  Apr 27, 2025, 6:03 PM
Y by
Find all sequences $a_n$ of positive integers such that

$a_{n+2}(a_{n+1} - k) = a_n(a_{n+1} + k)$

for all positive integers n.
This post has been edited 2 times. Last edited by Eternica, Apr 27, 2025, 6:50 PM
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dno1467
5 posts
dno1467
#2 h  Apr 27, 2025, 6:22 PM
Y by
Cute solution outline to a quite L problem:

The key step is to define the quantity $A_{n+1}$ such that $a_{n+2} = A_{n+1}(a_{n+1} + k)$.
Notice that this means $a_n = A_{n+1}(a_{n+1} - k)$

As a result, in order for $a_n$ and $a_{n+2}$ to be both integer, $A_{n+1}$'s denominator must divide $2k$, which in particular means that $A_n$ is bounded below

Now notice that $A_nA_{n+1}$ is bounded above (by using the definition of $A_i$ for both $n$ and $n+1$), which means that $A_i$ is bounded above too.

We may do some manipulation from here to obtain that $(A_n-1)k = (A_nA_{n-1}-1)(a_{n-1}+k).$
It's not hard to see that $a_i$ is unbounded, so from this and the boundedness of $A_i$ we immediately obtain that both sides are zero, so $A_n = 1$.

This means that $a_n = a_{n-1} + k$ for some n, so this relation holds for all n, and ultimately the characterisation of sequences are all linear sequences with common difference k.
This post has been edited 1 time. Last edited by dno1467, Apr 27, 2025, 6:23 PM
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