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, an hour ago
Reason: typo

inequalities

Functional Equation

by AnhQuang_67, Mar 26, 2025, 2:00 PM

Find all functions $f: \mathbb{N} \cup \{0\} \to \mathbb{N} \cup \{0\}$ satisfying: $f(f(m)+f(n))=m+n, \forall m, n \in \mathbb{N} \cup \{0\}$

function

algebra

functional equation

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, 6 hours ago

algebra

polynomial

integral points

by jhz, Mar 26, 2025, 1:14 AM

Prove: there exist integer $x_1,x_2,\cdots x_{10},y_1,y_2,\cdots y_{10}$ satisfying the following conditions:
$(1)$ $|x_i|,|y_i|\le 10^{10}$ for all $1\le i \le 10$
$(2)$ Define the set $S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},$ then $|S| = 1024$ ，and any rectangular strip of width 1 covers at most two points of S.

This post has been edited 1 time. Last edited by jhz, Today at 1:17 AM

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, an hour ago

geometry

Function on positive integers with two inputs

by Assassino9931, Jan 27, 2025, 10:03 AM

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$ . Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$ . Determine all possible values of $f(2025, 2025)$ .

This post has been edited 1 time. Last edited by Assassino9931, Jan 27, 2025, 10:06 AM

function

number theory

functional equation

SL 2015 G1: Prove that IJ=AH

by Problem_Penetrator, Jul 7, 2016, 6:31 PM

Let $ABC$ be an acute triangle with orthocenter $H$ . Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$ . Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$ . Prove that $IJ = AH$ .

This post has been edited 2 times. Last edited by v_Enhance, Jul 7, 2016, 7:46 PM
Reason: Use official SL wording

geometry

IMO Shortlist

Triangle

LMT (I know finally).

by Dr4gon39, Jun 25, 2016, 12:28 AM

So I'm really bored right now because you know, summer...

So I thought I'd finally post on LMT, as requested.

Forgive me y'all if my memory is kinda fuzzy.

LMT was my last ever middle school math competition.

So our school decided to send two teams, the A team consisting of Me, willwin4sure, dckx15, supermath7676 and redsoxfan. We also sent a B team but none of them really use AoPS so they're really quite irrelevant. I also tried to get cedric-the-reindeer to sign up but he/she didn't so whatever.

So we got there like, tooooooo early. We sat in the auditorium for a while and talked and took some pictures *cough* adults *cough*. Then our friends from flame math came and we made some small talk and whatnot. Then Clarke 1 came in, and they asked us where ghghghghghghghgh was. Benstien was there, but janabel (that's her username, I promise) wasn't because she was at like this orchestra thing. That's also why Doherty didn't send a team, so we didn't see OpalSeaDrag0n9 or ilikepie2003. But anyways it was kinda cool that Clarke 1 came to talk to us and flame math. (Sidenote on Clarke: Me and my friends have pretended that we're in a rivalry with Clarke ever since 6th grade, but in reality, we know we're rather insignificant in comparison to Jonas Clarke, so its kind of a quasi-pseudo-rivalry at most. But anyways, its been fun competing with you guys for the past three years, watching you guys blow us all away for the past three years. If you read this, best of luck in the future, and know that there's this little school in Andover vying to usurp your dominance one day (heheheheh just kidding, lets all be friends, am I right?))

But then the competition started, so we went into the testing room.

Individual round was hard. I think I only answered like 12 or 13 of the 20 (I think) questions.

But then came theme round.

Theme round was crazy...
One of the themes was about bijections, injections and surjections, and the explanation given wasn't exactly the best....
Also, all of the other problems seemed impossible too besides a few.... But yeah everyone did horribly on it.

Then came team round.
Me and willwin4sure went and did the proofs, but they were pretty hard. It was cool that I knew my proctor, because she did science bowl with ABCDE a few years back, but I promise there was no cheating. In the end we also worked on the potpourri with the other three guys, and we didn't do that good, but it wasn't that bad either. I just want to complain that they did put limits in the potpourri, but whatever, the problem wasn't hard.... Anyways, this group is usually pretty functional during team round.....

Then we had lunch. I had a lot of pizza. My friends know how much I eat. During lunch we all felt pretty terrible because we all thought that we did horribly during the rounds because all the questions were pretty hard, especially theme.... But anyways pizza was good.

Then came guts round. We got into the auditorium early, and me and dckx15 talked for a while with this LHS person who was running the competition about science bowl for a while which was pretty fun. Then we situated ourselves in the back of the auditorium and got ready for guts. The live scoreboard was pretty nerve racking. Of course, clarke 1 established a clear lead, and my team sort of floated around 3-6. Then we got down to the last two sets of problems and they shut down the scoreboard for "suspense reasons". The second to last set was extremely hard, because the problems were all related to each other, so we just put down 0 for all of them. Then we got to the last set, which was three estimation problems, that we did not get right.....

Then we waited for results.
I got 7th for individual round, which wasn't too bad (willwin4sure also got 8th). I got 10th for individual composite which wasn't bad at all considering my abysmal theme showing.... For team round we got 5th which also wasn't bad. Then guts round. OMIGOSH. We got 2nd. Crazy. Second Place. Just being Jonas. Which sort of makes it 1st place. Pretty cool if you ask me. Uhhhhh we also only got 9th for team score, which was slightly weird, but I guess its because half to eh team score is individual, so I guess some of our team members didn't perform up to standards on the individual rounds *hacking cough hacking cough*. But yeah the results weren't bad, especially with the 2nd place guts snag.

Overall, pretty good day if you ask me.

This post has been edited 1 time. Last edited by Dr4gon39, Jun 25, 2016, 12:31 AM

LMT

Prove that there exists a convex 1990-gon

by orl, Nov 11, 2005, 6:55 PM

Prove that there exists a convex 1990-gon with the following two properties :

a.) All angles are equal.
b.) The lengths of the 1990 sides are the numbers $1^2$ , $2^2$ , $3^2$ , $\cdots$ , $1990^2$ in some order.

This post has been edited 1 time. Last edited by orl, Aug 15, 2008, 4:18 PM

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 AnhQuang_67, Mar 26, 2025, 2:00 PM

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

by jhz, Mar 26, 2025, 1:14 AM

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

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

by Assassino9931, Jan 27, 2025, 10:03 AM

by Problem_Penetrator, Jul 7, 2016, 6:31 PM

by Dr4gon39, Jun 25, 2016, 12:28 AM

by orl, Nov 11, 2005, 6:55 PM

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