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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
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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As some nations like to say "Heavy theorems mostly do not help"
Assassino9931   9
N 28 minutes ago by EVKV
Source: European Mathematical Cup 2022, Senior Division, Problem 2
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.

a) Are there infinitely many lovely numbers?

b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?
9 replies
Assassino9931
Dec 20, 2022
EVKV
28 minutes ago
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congruence
moldovan   5
N an hour ago by EVKV
Source: Canada 2004
Let $p$ be an odd prime. Prove that:
\[\displaystyle\sum_{k=1}^{p-1}k^{2p-1} \equiv \frac{p(p+1)}{2} \pmod{p^2}\]
5 replies
moldovan
Jun 26, 2009
EVKV
an hour ago
J
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Checking a summand property for integers sufficiently large.
DinDean   1
N an hour ago by Double07
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 $a_1\mid a_2$, $a_2\mid a_3$, $\dots$, $a_{m-1}\mid a_m$.
1 reply
DinDean
2 hours ago
Double07
an hour ago
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Equations
Jackson0423   1
N an hour ago by Maxklark
Solve the system of equations
\[ \begin{cases} x - y z = 1,\\[2pt] y - z x = 2,\\[2pt] z - x y = 4. \end{cases} \]
1 reply
Jackson0423
3 hours ago
Maxklark
an hour ago
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Calculate the distance of chess king!!
egxa   3
N an hour ago by egxa
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.)
3 replies
egxa
Apr 18, 2025
egxa
an hour ago
J
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real+ FE
pomodor_ap   4
N an hour ago by jasperE3
Source: Own, PDC001-P7
Let $f : \mathbb{R}^+ \to \mathbb{R}^+$ be a function such that
$$f(x)f(x^2 + y f(y)) = f(x)f(y^2) + x^3$$for all $x, y \in \mathbb{R}^+$. Determine all such functions $f$.
4 replies
pomodor_ap
Yesterday at 11:24 AM
jasperE3
an hour ago
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FE solution too simple?
Yiyj1   8
N an hour 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
an hour ago
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Polynomials in Z[x]
BartSimpsons   16
N 2 hours 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
2 hours ago
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Why is the old one deleted?
EeEeRUT   13
N 2 hours 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
2 hours ago
J
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Factor sums of integers
Aopamy   2
N 2 hours 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
2 hours ago
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Least integer T_m such that m divides gauss sum
Al3jandro0000   33
N 2 hours 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
2 hours ago
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Estonian Math Competitions 2005/2006
STARS   2
N 2 hours 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
2 hours ago
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Sum of whose elements is divisible by p
nntrkien   43
N 3 hours 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
3 hours ago
J
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Arrangement of integers in a row with gcd
egxa   2
N 3 hours 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
3 hours ago
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J
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2019 Nepal National Mathematics Olympiad
Piinfinity   3
N Apr 4, 2025 by godchunguus
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
3 replies
Piinfinity
Oct 13, 2020
godchunguus
Apr 4, 2025
J
2019 Nepal National Mathematics Olympiad
G H J
Reveal topic tags
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algebra
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Piinfinity
61 posts
Piinfinity
#1 h  Oct 13, 2020, 5:56 AM • 2 Y
Y by Mango247, Mango247
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
Z K Y
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hyay
181 posts
hyay
#2 h  Oct 13, 2020, 8:59 AM
Y by
Solution
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Tintarn
9038 posts
Tintarn
#3 h  Oct 13, 2020, 10:44 AM
Y by
Piinfinity wrote:
Problem 31
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
$f(f(x))=x^2-x+1$
for all real numbers $x$. Determine $f(0)$.
This is Problem 5 from Baltic Way 2011. See here.
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godchunguus
15 posts
godchunguus
#4 h  Apr 4, 2025, 2:39 AM
Y by
$\text{Let} \ P(x)\text{ denote } f(f(x)) = x^2 - x + 1. \\ \\ P(1) \implies f(f(1)) = 1 \\ \\ P(f(1)) \implies f(f(f(1))) = f(1)^2 - f(1) + 1 \Rightarrow f(1) = f(1)^2 - f(1) + 1 \Rightarrow f(1)^2 - 2f(1) + 1 = 0 \Rightarrow f(1) = 1 \\ \\ P(0) \implies f(f(0)) = 1 \\ \\ P(f(0)) \implies f(f(f(0))) = f(0)^2 - f(0) + 1 \Rightarrow f(0)^2 - f(0) = 0 \Rightarrow f(0) = 0 \text{ or } 1. \\ \\ \text{Recall that } f(f(0)) = 1. \text{This would imply that } f(0) = 1\text{, as we also know that } f(1) = 1. \\ \\ \therefore f(0) = 1.$
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