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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 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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geometry
JetFire008   1
N 26 minutes ago by ohiorizzler1434
Given four concyclic points. For each subset of three points take the incenter. Show that the four incentres form a rectangle.
1 reply
JetFire008
Yesterday at 4:14 PM
ohiorizzler1434
26 minutes ago
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A pentagon inscribed in a circle of radius √2
tom-nowy   2
N 29 minutes ago by ohiorizzler1434
Can a pentagon with all rational side lengths be inscribed in a circle of radius $\sqrt{2}$ ?
2 replies
tom-nowy
2 hours ago
ohiorizzler1434
29 minutes ago
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BABBAGE'S THEOREM EXTENSION
Mathgloggers   2
N 31 minutes ago by Mathgloggers
A few days ago I came across. this interesting result is someone interested in proving this.

$\boxed{\sum_{k=1}^{p-1} \frac{1}{k} \equiv \sum_{k=p+1}^{2p-1} \frac{1}{k} \equiv \sum_{k=2p+1}^{3p-1}\frac{1}{k} \equiv.....\sum_{k=p(p-1)+1}^{p^2-1}\frac{1}{k} \equiv 0(mod p^2)}$
2 replies
Mathgloggers
Apr 29, 2025
Mathgloggers
31 minutes ago
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Parallelograms and concyclicity
Lukaluce   31
N 34 minutes ago by Ihatecombin
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
31 replies
Lukaluce
Apr 14, 2025
Ihatecombin
34 minutes ago
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My Unsolved FE in R+
ZeltaQN2008   2
N 41 minutes ago by megarnie
Source: Ho Chi Minh TST 2017 - 2018
Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all any $x,y\in (0,\infty):$
$$f(1+xf(y))=yf(x+y)$$
2 replies
ZeltaQN2008
2 hours ago
megarnie
41 minutes ago
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Something nice
KhuongTrang   32
N an hour ago by arqady
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
32 replies
KhuongTrang
Nov 1, 2023
arqady
an hour ago
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Infimum of decreasing sequence b_n/n^2
a1267ab   35
N an hour ago by shendrew7
Source: USA Winter TST for IMO 2020, Problem 1 and TST for EGMO 2020, Problem 3, by Carl Schildkraut and Milan Haiman
Choose positive integers $b_1, b_2, \dotsc$ satisfying
\[1=\frac{b_1}{1^2} > \frac{b_2}{2^2} > \frac{b_3}{3^2} > \frac{b_4}{4^2} > \dotsb\]and let $r$ denote the largest real number satisfying $\tfrac{b_n}{n^2} \geq r$ for all positive integers $n$. What are the possible values of $r$ across all possible choices of the sequence $(b_n)$?

Carl Schildkraut and Milan Haiman
35 replies
a1267ab
Dec 16, 2019
shendrew7
an hour ago
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IMO Genre Predictions
ohiorizzler1434   52
N 2 hours ago by justaguy_69
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
52 replies
ohiorizzler1434
May 3, 2025
justaguy_69
2 hours ago
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Inequalities
sqing   8
N 2 hours ago by sqing
Let $ a,b,c>0 $ and $ a+b\leq 16abc. $ Prove that
$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$Where $ k>0. $
$$ a+b+3c^3\geq\sqrt{\frac{2} {3}}$$$$ a+b+2c^4\geq \frac{5} {8}$$
8 replies
sqing
Sunday at 12:46 PM
sqing
2 hours ago
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\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2 }+\sqrt{c^2+a^2+2}\ge 6
parmenides51   19
N 3 hours ago by NicoN9
Source: JBMO Shortlist 2017 A1
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
19 replies
parmenides51
Jul 25, 2018
NicoN9
3 hours ago
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Inspired by Austria 2025
sqing   1
N 3 hours ago by sqing
Source: Own
Let $ a,b\geq 0 ,a,b\neq 1$ and $ a^2+b^2=1. $ Prove that$$ (a + b ) \left( \frac{a}{(b -1)^2} + \frac{b}{(a - 1)^2} \right) \geq 12+8\sqrt 2$$
1 reply
sqing
3 hours ago
sqing
3 hours ago
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Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   17
N 4 hours ago by Ilikeminecraft
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
17 replies
MarkBcc168
Apr 28, 2020
Ilikeminecraft
4 hours ago
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Inequality with a,b,c
GeoMorocco   7
N 5 hours ago by lele0305
Source: Morocco Training
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
7 replies
GeoMorocco
Apr 11, 2025
lele0305
5 hours ago
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Property of the divisors of k^3 - 2
Scilyse   2
N 6 hours ago by Assassino9931
Source: KoMaL A. 892
Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$. Show that there exists integers $a$, $b$, $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$.

Proposed by Csongor Beke and László Bence Simon, Cambridge
2 replies
Scilyse
Jan 13, 2025
Assassino9931
6 hours ago
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Theory of Equations
P162008   4
N Apr 25, 2025 by vanstraelen
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
4 replies
P162008
Apr 23, 2025
vanstraelen
Apr 25, 2025
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Theory of Equations
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P162008
182 posts
P162008
#1 h  Apr 23, 2025, 11:27 AM
Y by
Let $a,b,c,d$ and $e\in [-2,2]$ such that $\sum_{cyc} a = 0, \sum_{cyc} a^3 = 0, \sum_{cyc} a^5 = 10.$ Find the value of $\sum_{cyc} a^2.$
This post has been edited 2 times. Last edited by P162008, Apr 23, 2025, 11:28 AM
Reason: Typo
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vanstraelen
9010 posts
vanstraelen
#2 h  Apr 23, 2025, 7:48 PM • 1 Y
Y by P162008
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.
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P162008
182 posts
P162008
#3 h  Apr 23, 2025, 10:18 PM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

Yeah correct
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P162008
182 posts
P162008
#4 h  Apr 25, 2025, 7:46 AM
Y by
vanstraelen wrote:
Polynomial $P(x)=x^{5}-5x^{3}+5x-2$.

What was your approach?
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vanstraelen
9010 posts
vanstraelen
#5 h  Apr 25, 2025, 3:11 PM
Y by
$P(x)=x^{5}+p_{4}x^{4}+p_{3}x^{3}+p_{2}x^{2}+p_{1}x+p_{0}$.

$a$ is a root $\Rightarrow a^{5}+p_{4}a^{4}+p_{3}a^{3}+p_{2}a^{2}+p_{1}a+p_{0}=0$, etc.
$\sum_{cyc} a^5+p_{4} \cdot \sum_{cyc} a^4+p_{3} \cdot \sum_{cyc} a^3+p_{2} \cdot \sum_{cyc} a^2+p_{1} \sum_{cyc} a+5p_{0}=0$
or $10+p_{4} \cdot \sum_{cyc} a^4+p_{2} \cdot \sum_{cyc} a^2+5p_{0}=0$.
Let $p_{4}=p_{2}=0$, then $p_{0}=-2$.

New equation: $P(x)=x^{5}+px^{3}+qx-2$.
We want $x=2$ is a root, then $q=-15-4p$ and $P(x)=x^{5}+px^{3}-(15+4p)x-2$.
$P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]$.
We try $P(x)=(x-2)[x^{4}+2x^{3}+(p+4)x^{2}+(2p+8)x+1]=(x-2)(x^{2}+tx-1)^{2}$
and we find $t=1,p=-5$.
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