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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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no numbers of the form 80...01 are squares
Marius_Avion_De_Vanatoare   2
N 6 minutes ago by EVKV
Source: Moldova JTST 2024 P5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
2 replies
Marius_Avion_De_Vanatoare
Jun 10, 2024
EVKV
6 minutes ago
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f((x XOR f(y)) + y) = (f(x) XOR y) + y
the_universe6626   3
N 14 minutes 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
14 minutes ago
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2024 8's
Marius_Avion_De_Vanatoare   3
N 23 minutes 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
23 minutes ago
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pretty well known
dotscom26   0
23 minutes 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
23 minutes ago
0 replies
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Thanks u!
Ruji2018252   6
N 36 minutes 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
36 minutes ago
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Modular NT
oVlad   3
N 37 minutes 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
37 minutes ago
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Ratio conditions; prove angle XPA = angle AQY
MellowMelon   15
N 41 minutes 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
41 minutes ago
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IMO 2017 Problem 1
cjquines0   154
N an hour 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
an hour ago
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IMO 2018 Problem 5
orthocentre   76
N an hour 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
an hour ago
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Is this FE solvable?
Mathdreams   3
N an hour ago by jasperE3
Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]for all reals $x$ and $y$.
3 replies
Mathdreams
Tuesday at 6:58 PM
jasperE3
an hour ago
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Junior Balkan Mathematical Olympiad 2024- P3
Lukaluce   13
N an hour 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
an hour ago
J
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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
2 hours ago
Ianis
2 hours ago
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cursed tangent is xiooix
TestX01   2
N 3 hours ago by TestX01
Source: xiooix and i
Let $ABC$ be a triangle. Let $E$ and $F$ be the intersections of the $B$ and $C$ angle bisectors with the opposite sides. Let $S = (AEF) \cap (ABC)$. Let $W = SL \cap (AEF)$ where $L$ is the major $BC$ arc midpiont.
i)Show that points $S , B , C , W , E$ and $F$ are coconic on a conic $\mathcal{C}$
ii) If $\mathcal{C}$ intersects $(ABC)$ again at $T$, not equal to $B,C$ or $S$, then prove $AL$ and $ST$ concur on $(AEF)$

I will post solution in ~1 week if noone solves.
2 replies
TestX01
Feb 25, 2025
TestX01
3 hours ago
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Game on a row of 9 squares
EmersonSoriano   2
N 3 hours ago by Mr.Sharkman
Source: 2018 Peru TST Cono Sur P10
Let $n$ be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:

$(i)\:$ Choose a number of the form $2^j$, with $j$ a non-negative integer, and place it in an empty square.

$(ii)\:$ Choose two (not necessarily consecutive) squares containing the same number, say $2^j$. Replace the number in one of the squares with $2^{j+1}$ and erase the number in the other square.

At the end of the game, one square contains the number $2^n$, while the other squares are empty. Determine, as a function of $n$, the maximum number of turns Alex can make.
2 replies
EmersonSoriano
4 hours ago
Mr.Sharkman
3 hours ago
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J
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3 var inquality
sqing   1
N Mar 31, 2025 by sqing
Source: Own
Let $ a,b,c> 0 $ and $a+b+c=3. $ Prove that
$$ \frac{7}{a^2+b^2+c^2+a^2b^2+b^2c^2+c^2a^2 }+\frac{1}{abc} \geq\frac{13}{6}$$$$ \frac{43 }{a^2+b^2+c^2 }+\frac{10}{abc} \geq\frac{73}{3}$$$$ \frac{439 }{a^2+b^2+c^2 }+\frac{100}{abc} \geq\frac{739}{3}$$
1 reply
sqing
Mar 31, 2025
sqing
Mar 31, 2025
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3 var inquality
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Reveal topic tags
inequalities proposed
inequalities
algebra
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Source: Own
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sqing
41383 posts
sqing
#1 h  Mar 31, 2025, 4:19 AM
Y by
Let $ a,b,c> 0 $ and $a+b+c=3. $ Prove that
$$ \frac{7}{a^2+b^2+c^2+a^2b^2+b^2c^2+c^2a^2 }+\frac{1}{abc} \geq\frac{13}{6}$$$$ \frac{43 }{a^2+b^2+c^2 }+\frac{10}{abc} \geq\frac{73}{3}$$$$ \frac{439 }{a^2+b^2+c^2 }+\frac{100}{abc} \geq\frac{739}{3}$$
This post has been edited 1 time. Last edited by sqing, Mar 31, 2025, 4:49 AM
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sqing
41383 posts
sqing
#2 h  Mar 31, 2025, 4:58 AM
Y by
Let $ a,b,c> 0 $ and $a+b+c=3. $ Prove that
$$ \frac{34}{a^2+b^2+c^2+ab+bc+ca }+\frac{1}{abc} \geq\frac{20}{3}$$$$ \frac{347}{a^2+b^2+c^2+ab+bc+ca }+\frac{10}{abc} \geq\frac{407}{6}$$$$ \frac{3473}{a^2+b^2+c^2+ab+bc+ca }+\frac{100}{abc}\geq\frac{4073}{6}$$
This post has been edited 1 time. Last edited by sqing, Mar 31, 2025, 4:58 AM
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