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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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Be sure to mark your calendars for the following events:
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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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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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Simiplifying a Complicated Expression
phiReKaLk6781   5
N 29 minutes ago by P162008
Simplify: $ \frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-a)(b-c)}+\frac{c^3}{(c-a)(c-b)}$
5 replies
phiReKaLk6781
Mar 15, 2010
P162008
29 minutes ago
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2006 SMT Algebra #8 - Evaluate
bluecarneal   4
N 39 minutes ago by P162008
Evaluate:
\[\sum^{10}_{x=2} \dfrac{2}{x(x^2-1)}\]
.
4 replies
bluecarneal
Aug 24, 2011
P162008
39 minutes ago
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Σ to ∞
phiReKaLk6781   3
N 42 minutes ago by beyim
Evaluate: $ \sum\limits_{k=1}^\infty \frac{1}{k\sqrt{k+2}+(k+2)\sqrt{k}}$
3 replies
phiReKaLk6781
Mar 20, 2010
beyim
42 minutes ago
J
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Challenging Optimization Problem
Shiyul   2
N an hour ago by aidan0626
Let $xyz = 1$. Find the minimum and maximum values of $\frac{1}{1 + x + xy}$ + $\frac{1}{1 + y + yz}$ + $\frac{1}{1 + z + zx}$

Can anyone give me a hint? I got that either the minimum or maximum was 1, but I'm sure if I'm correct.
2 replies
Shiyul
5 hours ago
aidan0626
an hour ago
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Another System
worthawholebean   3
N 2 hours ago by P162008
Source: HMMT 2008 Guts Problem 33
Let $ a$, $ b$, $ c$ be nonzero real numbers such that $ a+b+c=0$ and $ a^3+b^3+c^3=a^5+b^5+c^5$. Find the value of
$ a^2+b^2+c^2$.
3 replies
worthawholebean
May 13, 2008
P162008
2 hours ago
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Inequality with three conditions
oVlad   2
N 2 hours ago by Quantum-Phantom
Source: Romania EGMO TST 2019 Day 1 P3
Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$
2 replies
oVlad
Yesterday at 1:48 PM
Quantum-Phantom
2 hours ago
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GCD Functional Equation
pinetree1   61
N 2 hours ago by ihategeo_1969
Source: USA TSTST 2019 Problem 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.

Ankan Bhattacharya
61 replies
pinetree1
Jun 25, 2019
ihategeo_1969
2 hours ago
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An easy FE
oVlad   3
N 3 hours ago by jasperE3
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
3 replies
1 viewing
oVlad
Yesterday at 1:36 PM
jasperE3
3 hours ago
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Interesting F.E
Jackson0423   12
N 3 hours ago by jasperE3
Show that there does not exist a function
\[ f : \mathbb{R}^+ \to \mathbb{R} \]satisfying the condition that for all \( x, y \in \mathbb{R}^+ \),
\[ f(x + y^2) \geq f(x) + y. \]

~Korea 2017 P7
12 replies
Jackson0423
Apr 18, 2025
jasperE3
3 hours ago
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p^3 divides (a + b)^p - a^p - b^p
62861   49
N 3 hours ago by Ilikeminecraft
Source: USA January TST for IMO 2017, Problem 3
Prove that there are infinitely many triples $(a, b, p)$ of positive integers with $p$ prime, $a < p$, and $b < p$, such that $(a + b)^p - a^p - b^p$ is a multiple of $p^3$.

Noam Elkies
49 replies
62861
Feb 23, 2017
Ilikeminecraft
3 hours ago
J
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3D geometry theorem
KAME06   0
3 hours ago
Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$. Prove that:
$$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$$
0 replies
KAME06
3 hours ago
0 replies
J
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Funny easy transcendental geo
qwerty123456asdfgzxcvb   1
N 3 hours ago by golue3120
Let $\mathcal{S}$ be a logarithmic spiral centered at the origin (ie curve satisfying for any point $X$ on it, line $OX$ makes a fixed angle with the tangent to $\mathcal{S}$ at $X$). Let $\mathcal{H}$ be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.

Prove that for a point $P$ on the spiral, the polar of $P$ wrt. $\mathcal{H}$ is tangent to the spiral.
1 reply
qwerty123456asdfgzxcvb
6 hours ago
golue3120
3 hours ago
J
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domino question
kjhgyuio   0
3 hours ago
........
0 replies
kjhgyuio
3 hours ago
0 replies
J
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demonic monic polynomial problem
iStud   0
3 hours ago
Source: Monthly Contest KTOM April P4 Essay
(a) Let $P(x)$ be a monic polynomial so that there exists another real coefficients $Q(x)$ that satisfy
\[P(x^2-2)=P(x)Q(x)\]Determine all complex roots that are possible from $P(x)$
(b) For arbitrary polynomial $P(x)$ that satisfies (a), determine whether $P(x)$ should have real coefficients or not.
0 replies
iStud
3 hours ago
0 replies
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Number of solutions
Ecrin_eren   4
N Apr 18, 2025 by Ecrin_eren
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.

4 replies
Ecrin_eren
Apr 3, 2025
Ecrin_eren
Apr 18, 2025
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Number of solutions
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Reveal topic tags
number theory
integer polynomials
Simon favorite factoring trick
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Ecrin_eren
56 posts
Ecrin_eren
#1 h  Apr 3, 2025, 11:27 AM
Y by
The given equation is:

x³ + 4y³ + 2y = (2024 + 2y)(xy + 1)

The question asks for the number of integer solutions.
Z K Y
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jjsunpu
130 posts
jjsunpu
#2 h  Apr 3, 2025, 3:54 PM
Y by
for x or y?
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Ecrin_eren
56 posts
Ecrin_eren
#3 h  Apr 3, 2025, 6:41 PM
Y by
Number of (x,y) pairs
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rchokler
2965 posts
rchokler
#5 h  Apr 3, 2025, 8:00 PM
Y by
$x^3+4y^3+2y=(2024+2y)(xy+1)\implies x^3-2xy^2+4y^3-2024xy-2024=0\implies$

Putting $z=\frac{x}{y}$ for $y\neq 0$ gives $z^3-2z+4-\frac{2024z}{y}-\frac{1}{y^3}=0$ Letting $|y|\to\infty$ results in $z^3-2z+4=0\implies(z+2)(z^2-2z+2)=0\implies z=-2$.

So we have an asymptote $x+2y=c$.

So plug in $x=c-2y$ to get $(c-2y)^3+4y^3+2y=(2024+2y)((c-2y)y + 1)\implies (10c+4048)y^2-(6c^2+2024c)y+c^3-2024=0$.

Therefore, $y=\frac{3c^2+1012c\pm \sqrt{(3c^2+1012c)^2-(10c+4048)(c^3-2024)}}{10c+4048}=\frac{3c^2+1012c\pm\sqrt{-c^4 + 2024 c^3 + 1024144 c^2 + 20240 c + 8193152}}{10c+4048}$.

The integer range is $-419\leq c\leq 2443$. The discriminant is never a square.

There are no integer solutions for $x^3+4y^3+2y=(2024+2y)(xy+1)$.
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Ecrin_eren
56 posts
Ecrin_eren
#6 h  Apr 18, 2025, 3:19 PM
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Any other solutions ?
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