cyclic ineq not tight

by RainbowNeos, Mar 26, 2025, 2:24 PM

Given $n\geq 3$ and $x_i\geq 0, 1\leq i\leq n$ with sum $1$ . Show that
$\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.$ where $x_{n+1}=x_1$ .

This post has been edited 1 time. Last edited by RainbowNeos, 24 minutes ago
Reason: typo

inequalities

function

by MuradSafarli, Mar 26, 2025, 1:26 PM

Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying the equation for all real numbers $x, y$ :
$f(x^2 + y + f(y)) = f(x)^2$

function

algebra

Long polynomial factorization

by wassupevery1, Mar 26, 2025, 7:33 AM

For each prime $p$ of the form $4k+3$ with $k \in \mathbb{Z}^+$ , consider the polynomial $Q(x)=px^{2p} - x^{2p-1} + p^2x^{\frac{3p+1}{2}} - px^{p+1} +2(p^2+1)x^p -px^{p-1}+ p^2 x^{\frac{p-1}{2}} -x + p.$ Determine all ordered pairs of polynomials $f, g$ with integer coefficients such that $Q(x)=f(x)g(x)$ .

This post has been edited 1 time. Last edited by wassupevery1, 5 hours ago

algebra

polynomial

God do bosses have a hard job

by AshAuktober, Mar 25, 2025, 4:36 PM

The boss has to assign ten job positions to ten candidates, considering two parameters: preference and ability. If candidate A prefers job $v$ to job $u$ and has a better ability in job $v$ than candidate B, but A is assigned job $u$ and B is assigned job $v$ , then A will complain. Also, if it is possible to assign each job to a candidate with a higher ability, the director will complain. Show that the boss can assign the jobs so as to avoid any complaints.

combinatorics

Parallel lines in an acute triangle

by buratinogigle, Mar 25, 2025, 9:01 AM

Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$ . Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$ , respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$ , respectively. Let $S$ be the circumcenter of triangle $A'B'C'$ , and let $H'$ be the orthocenter of triangle $DEF$ . Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$ . Prove that $HJ$ is parallel to $H'S$ .

geometry

Circles and Chords

by steven_zhang123, Mar 23, 2025, 7:29 AM

(1) Let $A$ , $B$ and $C$ be points on circle $O$ divided into three equal parts. Construct three equal circles $O_1$ , $O_2$ , and $O_3$ tangent to $O$ internally at points $A$ , $B$ , and $C$ respectively. Let $P$ be any point on arc $AC$ , and draw tangents $PD$ , $PE$ , and $PF$ to circles $O_1$ , $O_2$ , and $O_3$ respectively. Prove that $PE = PD + PF$ .

(2) Let $A_1$ , $A_2$ , $\cdots$ , $A_n$ be points on circle $O$ divided into $n$ equal parts. Construct $n$ equal circles $O_1$ , $O_2$ , $\cdots$ , $O_n$ tangent to $O$ internally at $A_1$ , $A_2$ , $\cdots$ , $A_n$ . Let $P$ be any point on circle $O$ , and draw tangents $PB_1$ , $PB_2$ , $\cdots$ , $PB_n$ to circles $O_1$ , $O_2$ , $\cdots$ , $O_n$ . If the sum of $k$ of $PB_1$ , $PB_2$ , $\cdots$ , $PB_n$ equals the sum of the remaining $n-k$ (where $n \geq k \geq 1$ ), find all such $n$ .

Attachments:

This post has been edited 1 time. Last edited by steven_zhang123, 19 minutes ago

geometry

Something nice

by KhuongTrang, Nov 1, 2023, 12:56 PM

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$

This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM

inequalities

On Hamilton and History

by Dr4gon39, Jun 25, 2016, 12:44 AM

So besides math the other subject that I like the most is definitely history...

In fact I probably like school history at least a teensy more than school math....

Competition math definitely better than school history though....

But anyways, Hamilton is really really cool. Really.

I know that some people don't get it, like, why would you want to listen to someone rap about a bastard, orphan, son of a whore and a scotsman.., while you could like, listen to eminem throw lyrics at you at supersonic speed. But anyways, Hamilton appeals to me. A lot. I can't even pinpoint why. Perhaps its just my love for history. Dunno. But Hamilton is awesome and that is the point of this blog post. I recommend you all take the chance and listen to the entire album in the right order at least once, (thanks for recommending that to me, you know who you are *winks*).

And why do I like history?

I do know that... like... unlike Hamilton.... there is a specific reason to that.
So History is this interesting thing. It repeats itself. In fact, I even started all my 6th grade history essays with something that looked like "through the cycles of history", and my teacher (who I credit for making me love history) gobbled it all up. But yeah, history does repeat itself. It teaches us about what goes on now, and it teachers us where all the greats made mistakes. But most importantly, from all the mistakes the greats made, we learn from them. And because of that, we become better people.

Love Hamilton. Love History.

hamilton

history

American History

Solution of a cubic

by Rushil, Oct 10, 2005, 4:53 AM

Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$ . prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$

algebra

polynomial

algebra unsolved

IMO ShortList 2003, number theory problem 1

by orl, Oct 4, 2004, 9:29 PM

Let $m$ be a fixed integer greater than $1$ . The sequence $x_0$ , $x_1$ , $x_2$ , $\ldots$ is defined as follows:
$x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .

Proposed by Marcin Kuczma, Poland

Attachments:

This post has been edited 1 time. Last edited by djmathman, May 27, 2018, 3:50 PM
Reason: changed display according to https://anhngq.files.wordpress.com/2010/07/imo-2003-shortlist.pdf

cyclic sum 1 / (a(b+1)) + ... >= 3 / (1+abc)

by Arne, Aug 17, 2003, 8:56 AM

Let $a$ , $b$ , $c$ be positive real numbers. Prove the inequality
$\frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}.$

inequalities

rearrangement inequality

inequalities theorems

inequalities proposed

Submit

ooh hello
by jj_ca888, Nov 18, 2019, 10:36 PM
it's been a good ride.
by ghghghghghghghgh, Jul 30, 2019, 2:17 AM
Vincent's Fan
by Supermath7676, Feb 14, 2017, 11:48 PM
Clearly just a function convolution.
by willwin4sure, Nov 28, 2016, 5:11 PM
yvincent
by ghghghghghghghgh, Nov 28, 2016, 1:22 AM
oh dearr..
by stephy2003, Oct 12, 2016, 4:55 PM
Lol @ghghghghghghgh when I delete your comment and the admins refer to it as taking out the "trash"
Touche
by Dr4gon39, Oct 11, 2016, 9:18 PM
Vincent's Fan
by Supermath7676, Oct 11, 2016, 8:13 PM
China......
by willwin4sure, Aug 4, 2016, 11:34 PM
all the many facets of me
by RedSoxFan, Jul 20, 2016, 10:21 PM
We need a chocolate bar
by stephy2003, Jun 23, 2016, 7:17 PM
Post on LMT?
by willwin4sure, Apr 20, 2016, 8:53 PM
King of kings. XD I thought you were the doctor of sides Dr. Gon.

@dckx because /Wilma/ didn't elaborate with instructions. I is creary veri clueless
by stephy2003, Apr 17, 2016, 3:27 AM
You lost your chance...
by dckx15, Apr 16, 2016, 9:24 PM
AHEM.
BEHOLD THE KING!
THE KING OF KINGS!
by Dr4gon39, Apr 16, 2016, 1:45 PM
~~He's coming back soon, do we want to kill his blog or not?~~
by stephy2003, Apr 15, 2016, 12:13 AM
You're very welcome
by willwin4sure, Apr 13, 2016, 9:22 PM
~~Thanks for knowing my name~~
by stephy2003, Apr 13, 2016, 2:07 PM
Stephen you can post right?
by willwin4sure, Apr 13, 2016, 12:23 PM
~~Guys his in D.C. this week, let's kill his blog~~
by stephy2003, Apr 12, 2016, 11:44 PM
Are you not going to talk about LMT?
by dckx15, Apr 11, 2016, 4:13 PM
Lolololol RedSoxFan just rekt your proof
by willwin4sure, Apr 5, 2016, 10:50 PM
Yay you fixed the typos
by RedSoxFan, Apr 5, 2016, 10:15 PM
@willwin4sure, i dunno, is that allowed?
by Dr4gon39, Apr 5, 2016, 8:44 PM
No Mr. Gon, willwin4sure is
by stephy2003, Apr 5, 2016, 8:14 PM
Can't you just cross multiply to determine if a/b > c/d then ad>bc?
by willwin4sure, Apr 5, 2016, 5:22 PM
You have too many typos
by RedSoxFan, Apr 5, 2016, 5:13 PM
#**********CLUBTAKEOVER.
Harvard is next.
by willwin4sure, Apr 5, 2016, 11:58 AM
I'm obnoxious?
by Dr4gon39, Apr 5, 2016, 11:03 AM
yes he was
and to me
and redsoxfan
by ilikepie2003, Apr 5, 2016, 12:46 AM
Were you obnoxious to Emc?
by stephy2003, Apr 5, 2016, 12:41 AM
Post about your amazing MagMar experience.
by willwin4sure, Apr 4, 2016, 10:10 PM
~~a phat lizard~~ someone who can post, but usually doesn't. I kind of just use it to bookmark blogs. amd this is why summitwei never gets any views from me
by stephy2003, Mar 16, 2016, 2:58 AM
Dunno. HWAT is a contrib?
by Dr4gon39, Mar 16, 2016, 2:31 AM
Can I be a contrib? Thx!
by ilikepie2003, Mar 4, 2016, 3:59 AM
Good luck on the AIME today!
by stephy2003, Mar 3, 2016, 11:05 PM
Well I guess the only thing different are the amount of h's in UGHHHHHH
by ilikepie2003, Feb 29, 2016, 12:12 AM
Agreed @stephy
by dckx15, Feb 28, 2016, 3:54 PM
Barely refrains from saying real name. *@Dr4gon39*, you have the most creative post names, I mean they totally aren't annoyingly repetitive
by stephy2003, Feb 24, 2016, 12:51 AM
ALLO
C'EST MOI
by ilikepie2003, Feb 20, 2016, 10:30 PM
Second Shout.
by stephy2003, Feb 19, 2016, 3:41 PM
first shout >
by doitsudoitsu, Feb 17, 2016, 12:17 AM

42 shouts

My blog. Duh.

by RainbowNeos, Mar 26, 2025, 2:24 PM

by MuradSafarli, Mar 26, 2025, 1:26 PM

by wassupevery1, Mar 26, 2025, 7:33 AM

by AshAuktober, Mar 25, 2025, 4:36 PM

by buratinogigle, Mar 25, 2025, 9:01 AM

by steven_zhang123, Mar 23, 2025, 7:29 AM

by KhuongTrang, Nov 1, 2023, 12:56 PM

by Dr4gon39, Jun 25, 2016, 12:44 AM

by Rushil, Oct 10, 2005, 4:53 AM

by orl, Oct 4, 2004, 9:29 PM

by Arne, Aug 17, 2003, 8:56 AM