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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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
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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trolling geometry problem
iStud   1
N 10 minutes ago by iStud
Source: Monthly Contest KTOM April P3 Essay
Given a cyclic quadrilateral $ABCD$ with $BC<AD$ and $CD<AB$. Lines $BC$ and $AD$ intersect at $X$, and lines $CD$ and $AB$ intersect at $Y$. Let $E,F,G,H$ be the midpoints of sides $AB,BC,CD,DA$, respectively. Let $S$ and $T$ be points on segment $EG$ and $FH$, respectively, so that $XS$ is the angle bisector of $\angle{DXA}$ and $YT$ is the angle bisector of $\angle{DYA}$. Prove that $TS$ is parallel to $BD$ if and only if $AC$ divides $ABCD$ into two triangles with equal area.
1 reply
iStud
3 hours ago
iStud
10 minutes ago
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basically INAMO 2010/6
iStud   3
N 10 minutes ago by iStud
Source: Monthly Contest KTOM April P1 Essay
Call $n$ kawaii if it satisfies $d(n)+\varphi(n)=n+1$ ($d(n)$ is the number of positive factors of $n$, while $\varphi(n)$ is the number of integers not more than $n$ that are relatively prime with $n$). Find all $n$ that is kawaii.
3 replies
iStud
3 hours ago
iStud
10 minutes ago
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GCD of a sequence
oVlad   7
N 30 minutes ago by grupyorum
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$
7 replies
oVlad
Yesterday at 1:35 PM
grupyorum
30 minutes ago
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Another System
worthawholebean   3
N 36 minutes 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
36 minutes ago
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Inequality with three conditions
oVlad   2
N an hour 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
an hour ago
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GCD Functional Equation
pinetree1   61
N an hour 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
an hour ago
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An easy FE
oVlad   3
N an hour 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
oVlad
Yesterday at 1:36 PM
jasperE3
an hour ago
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Interesting F.E
Jackson0423   12
N an hour 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
an hour ago
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p^3 divides (a + b)^p - a^p - b^p
62861   49
N 2 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
1 viewing
62861
Feb 23, 2017
Ilikeminecraft
2 hours ago
J
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3D geometry theorem
KAME06   0
2 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
2 hours ago
0 replies
J
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Funny easy transcendental geo
qwerty123456asdfgzxcvb   1
N 2 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
5 hours ago
golue3120
2 hours ago
J
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domino question
kjhgyuio   0
2 hours ago
........
0 replies
kjhgyuio
2 hours ago
0 replies
J
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demonic monic polynomial problem
iStud   0
2 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
2 hours ago
0 replies
J
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fun set problem
iStud   0
2 hours ago
Source: Monthly Contest KTOM April P2 Essay
Given a set $S$ with exactly 9 elements that is subset of $\{1,2,\dots,72\}$. Prove that there exist two subsets $A$ and $B$ that satisfy the following:
- $A$ and $B$ are non-empty subsets from $S$,
- the sum of all elements in each of $A$ and $B$ are equal, and
- $A\cap B$ is an empty subset.
0 replies
iStud
2 hours ago
0 replies
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Ratio of projections of a tetrahedron on two planes.
Goutham   0
Mar 17, 2011
Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$
0 replies
Goutham
Mar 17, 2011
0 replies
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Ratio of projections of a tetrahedron on two planes.
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Goutham
3130 posts
Goutham
#1 h  Mar 17, 2011, 5:45 PM • 2 Y
Y by Adventure10, Mango247
Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval $[\frac{1}{\sqrt{2}}, \sqrt{2}].$
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