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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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
Yesterday at 3:18 PM
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((x XOR f(y)) + y) = (f(x) XOR y) + y
the_universe6626   3
N an hour ago by jasperE3
Source: Janson MO 5 P4
Find all functions $f:\mathbb{Z}_{\ge0}\rightarrow\mathbb{Z}_{\ge0}$ such that
\[f((x\oplus f(y))+y)=(f(x)\oplus y)+y\]Note: $\oplus$ denotes the bitwise XOR operation. For example, $1001_2 \oplus 101_2 = 1100_2$.

(Proposed by ja.)
3 replies
the_universe6626
Feb 21, 2025
jasperE3
an hour ago
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2024 8's
Marius_Avion_De_Vanatoare   3
N an hour ago by EVKV
Source: Moldova JTST 2024 P2
Prove that the number $ \underbrace{88\dots8}_\text{2024\; \textrm{times}}$ is divisible by 2024.
3 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
an hour ago
J
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pretty well known
dotscom26   0
an hour ago
Let $\triangle ABC$ be a scalene triangle such that $\Omega$ is its incircle. $AB$ is tangent to $\Omega$ at $D$. A point $E$ ($E \notin \Omega$) is located on $BC$.

Let $\omega_1$, $\omega_2$, and $\omega_3$ be the incircles of the triangles $BED$, $ADE$, and $AEC$, respectively.

Show that the common tangent to $\omega_1$ and $\omega_3$ is also tangent to $\omega_2$.

0 replies
dotscom26
an hour ago
0 replies
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Thanks u!
Ruji2018252   6
N an hour ago by jasperE3
Find all $f:\mathbb{R}\to\mathbb{R}$ and
\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]
6 replies
Ruji2018252
Mar 26, 2025
jasperE3
an hour ago
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Modular NT
oVlad   3
N an hour ago by EVKV
Source: Romania JBMO TST 2024 Day 1 P1
Find all the positive integers $a{}$ and $b{}$ such that $(7^a-5^b)/8$ is a prime number.

Cosmin Manea and Dragoș Petrică
3 replies
oVlad
Jul 31, 2024
EVKV
an hour ago
J
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Ratio conditions; prove angle XPA = angle AQY
MellowMelon   15
N an hour ago by cj13609517288
Source: USA TSTST 2011/2012 P2
Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.
15 replies
MellowMelon
Jul 26, 2011
cj13609517288
an hour ago
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IMO 2017 Problem 1
cjquines0   154
N 2 hours ago by blueprimes
Source: IMO 2017 Problem 1
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} = \begin{cases} \sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\ a_n + 3 & \text{otherwise.} \end{cases} $$Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitely many values of $n$.

Proposed by Stephan Wagner, South Africa
154 replies
cjquines0
Jul 18, 2017
blueprimes
2 hours ago
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IMO 2018 Problem 5
orthocentre   76
N 2 hours ago by Maximilian113
Source: IMO 2018
Let $a_1$, $a_2$, $\ldots$ be an infinite sequence of positive integers. Suppose that there is an integer $N > 1$ such that, for each $n \geq N$, the number
$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \cdots + \frac{a_{n-1}}{a_n} + \frac{a_n}{a_1}$$is an integer. Prove that there is a positive integer $M$ such that $a_m = a_{m+1}$ for all $m \geq M$.

Proposed by Bayarmagnai Gombodorj, Mongolia
76 replies
orthocentre
Jul 10, 2018
Maximilian113
2 hours ago
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Junior Balkan Mathematical Olympiad 2024- P3
Lukaluce   13
N 2 hours ago by EVKV
Source: JBMO 2024
Find all triples of positive integers $(x, y, z)$ that satisfy the equation

$$2020^x + 2^y = 2024^z.$$
Proposed by Ognjen Tešić, Serbia
13 replies
Lukaluce
Jun 27, 2024
EVKV
2 hours ago
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School Math Problem
math_cool123   3
N 2 hours ago by jkim0656
Find all ordered pairs of nonzero integers $(a, b)$ that satisfy $$(a^2+b)(a+b^2)=(a-b)^3.$$
3 replies
math_cool123
Yesterday at 5:03 AM
jkim0656
2 hours ago
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Geometry :3c
popop614   2
N 2 hours ago by Ianis
Source: MINE :<
Quadrilateral $ABCD$ has an incenter $I$ Suppose $AB > BC$. Let $M$ be the midpoint of $AC$. Suppose that $MI \perp BI$. $DI$ meets $(BDM)$ again at point $T$. Let points $P$ and $Q$ be such that $T$ is the midpoint of $MP$ and $I$ is the midpoint of $MQ$. Point $S$ lies on the plane such that $AMSQ$ is a parallelogram, and suppose the angle bisectors of $MCQ$ and $MSQ$ concur on $IM$.

The angle bisectors of $\angle PAQ$ and $\angle PCQ$ meet $PQ$ at $X$ and $Y$. Prove that $PX = QY$.
2 replies
popop614
3 hours ago
Ianis
2 hours ago
J
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Real variables inequality
JK1603JK   0
3 hours ago
Let a,b,c\in R then prove that \frac{15}{2}\cdot\frac{a^2+b^2+c^2}{(a+b+c)^2}+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}\ge 4
0 replies
1 viewing
JK1603JK
3 hours ago
0 replies
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2024 COMC B1
QueenArwen   2
N 4 hours ago by EVKV
For any positive integer number $k$, the factorial $k!$ is defined as a product of all integers between $1$ and $k$ inclusive: $k!=k\times{(k-1)}\times\dots\times{1}$.
Let $s(n)$ denote the sum of the first $n$ factorials, i.e.
$$s(n)=\underbrace{n\times{(n-1)}\times\dots\times{1}}_{n!}+\underbrace{(n-1)\times{(n-2)}\times\dots\times{1}}_{(n-1)!}+\cdots +\underbrace{2\times{1}}_{2!}+\underbrace{1}_{1!}$$Find the remainder when $s(2024)$ is divided by $8$
2 replies
QueenArwen
Nov 4, 2024
EVKV
4 hours ago
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Easy FE; source unknown
NamelyOrange   3
N 6 hours ago by Mathdreams
Find (with proof) all $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(x)\ge x$ and $f(f(x)) = x$.
3 replies
NamelyOrange
Yesterday at 8:48 PM
Mathdreams
6 hours ago
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Prove that
abduqahhor_math   3
N Yesterday at 5:52 PM by rchokler
n^2+3n+5 is not divided to 121
3 replies
abduqahhor_math
Mar 31, 2025
rchokler
Yesterday at 5:52 PM
J
Prove that
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Reveal topic tags
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abduqahhor_math
5 posts
abduqahhor_math
#1 h  Mar 31, 2025, 5:37 PM
Y by
n^2+3n+5 is not divided to 121
This post has been edited 1 time. Last edited by abduqahhor_math, Mar 31, 2025, 5:38 PM
Reason: ! is extra
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Kempu33334
593 posts
Kempu33334
#2 h  Mar 31, 2025, 6:01 PM
Y by
abduqahhor_math wrote:
n^2+3n+5 is not divided to 121

Is the problem:

Prove that \[n^2+3n+5 \equiv 0\pmod{121}\]has no solutions?
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abduqahhor_math
5 posts
abduqahhor_math
#3 h  Yesterday at 4:14 PM
Y by
yes it is
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rchokler
2947 posts
rchokler
#4 h  Yesterday at 5:52 PM
Y by
First solve $n^2+3n+5\equiv 0\mod 11$.

\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c} $n$&0&1&2&3&4&5&6&7&8&9&10\\ \hline $n^2+3n+5$&5&9&4&1&0&1&4&9&5&3&3 \end{tabular}
So we have $n\equiv 4\mod 11$.

Next plug in $n=11k+4$.

$n^2+3n+5=(11k+4)^2+3(11k+4)+5=121k^2+88k+16+33k+12+5=121k^2+121k+33\equiv 33\mod 121$

$\square$
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