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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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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
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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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FE solution too simple?
Yiyj1   8
N a few seconds ago by lksb
Source: 101 Algebra Problems from the AMSP
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the equality $$f(f(x)+y) = f(x^2-y)+4f(x)y$$holds for all pairs of real numbers $(x,y)$.

My solution

I feel like my solution is too simple. Is there something I did wrong or something I missed?
8 replies
Yiyj1
Apr 9, 2025
lksb
a few seconds ago
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Polynomials in Z[x]
BartSimpsons   16
N 10 minutes ago by bin_sherlo
Source: European Mathematical Cup 2017 Problem 4
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$is a square of an integer for all nonnegative integers $n, m$.

Remark: For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$.

Proposed by Adrian Beker.
16 replies
BartSimpsons
Dec 27, 2017
bin_sherlo
10 minutes ago
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Why is the old one deleted?
EeEeRUT   13
N 26 minutes ago by EVKV
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
13 replies
EeEeRUT
Apr 16, 2025
EVKV
26 minutes ago
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Calculate the distance of chess king!!
egxa   2
N 31 minutes ago by Turkish_sniper
Source: All Russian 2025 9.4
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
2 replies
egxa
Apr 18, 2025
Turkish_sniper
31 minutes ago
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Checking a summand property for integers sufficiently large.
DinDean   0
41 minutes ago
For any fixed integer $m\geqslant 2$, prove that there exists a positive integer $f(m)$, such that for any integer $n\geqslant f(m)$, $n$ can be expressed by a sum of positive integers $a_i$'s as
\[n=a_1+a_2+\dots+a_m,\]where $a_1\mid a_2$, $a_2\mid a_3$, $\dots$, $a_{m-1}\mid a_m$.
0 replies
DinDean
41 minutes ago
0 replies
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Factor sums of integers
Aopamy   2
N an hour ago by cadaeibf
Let $n$ be a positive integer. A positive integer $k$ is called a benefactor of $n$ if the positive divisors of $k$ can be partitioned into two sets $A$ and $B$ such that $n$ is equal to the sum of elements in $A$ minus the sum of the elements in $B$. Note that $A$ or $B$ could be empty, and that the sum of the elements of the empty set is $0$.

For example, $15$ is a benefactor of $18$ because $1+5+15-3=18$.

Show that every positive integer $n$ has at least $2023$ benefactors.
2 replies
Aopamy
Feb 23, 2023
cadaeibf
an hour ago
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Least integer T_m such that m divides gauss sum
Al3jandro0000   33
N an hour ago by NerdyNashville
Source: 2020 Iberoamerican P2
Let $T_n$ denotes the least natural such that
$$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$Find all naturals $m$ such that $m\ge T_m$.

Proposed by Nicolás De la Hoz
33 replies
Al3jandro0000
Nov 17, 2020
NerdyNashville
an hour ago
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Estonian Math Competitions 2005/2006
STARS   2
N an hour ago by jasperE3
Source: Juniors Problem 4
A $ 9 \times 9$ square is divided into unit squares. Is it possible to fill each unit square with a number $ 1, 2,..., 9$ in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
2 replies
STARS
Jul 30, 2008
jasperE3
an hour ago
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Sum of whose elements is divisible by p
nntrkien   43
N an hour ago by lpieleanu
Source: IMO 1995, Problem 6, Day 2, IMO Shortlist 1995, N6
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?
43 replies
nntrkien
Aug 8, 2004
lpieleanu
an hour ago
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Arrangement of integers in a row with gcd
egxa   2
N an hour ago by Qing-Cloud
Source: All Russian 2025 10.5 and 11.5
Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained?
2 replies
egxa
Apr 18, 2025
Qing-Cloud
an hour ago
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Integer representation
RL_parkgong_0106   1
N an hour ago by Jackson0423
Source: Own
Show that for any positive integer $n$, there exists some positive integer $k$ that makes the following equation have no integer root $(x_1, x_2, x_3, \dots, x_n)$.

$$x_1^{2^1}+x_2^{2^2}+x_3^{2^3}+\dots+x_n^{2^n}=k$$
1 reply
RL_parkgong_0106
4 hours ago
Jackson0423
an hour ago
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Equations
Jackson0423   0
an hour ago
Solve the system of equations
\[ \begin{cases} x - y z = 1,\\[2pt] y - z x = 2,\\[2pt] z - x y = 4. \end{cases} \]
0 replies
Jackson0423
an hour ago
0 replies
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Factor of P(x)
Brut3Forc3   19
N an hour ago by xytunghoanh
Source: 1976 USAMO Problem 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\] prove that $ x-1$ is a factor of $ P(x)$.
19 replies
Brut3Forc3
Apr 4, 2010
xytunghoanh
an hour ago
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2^x+3^x = yx^2
truongphatt2668   1
N an hour ago by Jackson0423
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
1 reply
truongphatt2668
2 hours ago
Jackson0423
an hour ago
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J
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Tetrahedron with circumcircles having the same radius
Omid Hatami   2
N Jan 24, 2009 by mihai miculita
Source: Iranian National Olympiad (3rd Round) 2003
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB=CD$, $ AC=BD$ and $ AD=BC$.
2 replies
Omid Hatami
Jan 24, 2009
mihai miculita
Jan 24, 2009
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Tetrahedron with circumcircles having the same radius
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Reveal topic tags
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Source: Iranian National Olympiad (3rd Round) 2003
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Omid Hatami
1275 posts
Omid Hatami
#1 h  Jan 24, 2009, 1:58 AM • 2 Y
Y by Adventure10, Mango247
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB=CD$, $ AC=BD$ and $ AD=BC$.
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Kunihiko_Chikaya
14512 posts
Kunihiko_Chikaya
#2 h  Jan 24, 2009, 2:24 AM • 2 Y
Y by Adventure10, Mango247
We can make the tetrahedron by 6 diagonal of the attached rectangular parallelepiped.
Attachments:
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mihai miculita
666 posts
mihai miculita
#3 h  Jan 24, 2009, 10:25 AM • 2 Y
Y by Adventure10, Mango247
$ \mbox{Let } O_a, O_b, O_c, O_d \mbox{ center of circumcircle of } \triangle{BCD}, \triangle{ACD}, \triangle{ABC}; \mbox{ and } R_a, R_b, R_c, R_d \mbox{ circumradius this circles.}$
$ \mbox{If O is center of circumscribed sphere and I center of inscribed sphere of tetrahedron } ABCD, \mbox{and } \\ R_a = R_b = R_c = R_d, \mbox{ then: } |OO_a| = |OO_b| = |OO_c| = |OO_d|\Rightarrow O\equiv{I} \Rightarrow \\ \Rightarrow ABCD - \mbox{isoscele tetrahedron}\Rightarrow |AB| = |BD|, |AC| = |BD|, |AD| = |BC|.$
view: Nathan Altshiller-Court: "MODERN PURE SOLID GEOMETRY"(second edition); p.107; 304 Theorem.
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