Problem 12

by SlovEcience, Jun 2, 2025, 3:46 AM

Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that
\[ f(x^4 + 5y^4 + 10z^4) = f(x)^4 + 5f(y)^4 + 10f(z)^4 \]for all \( x, y, z \in \mathbb{N} \).
Function equations
algebra
function
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Channel name changed

by Plane_geometry_youtuber, Jun 1, 2025, 9:31 PM

Hi,

Due to the search handle issue in youtube. My channel is renamed to Olympiad Geometry Club. And the new link is as following:

https://www.youtube.com/@OlympiadGeometryClub

Recently I introduced the concept of harmonic bundle. I will move on to the conjugate median soon. In the future, I will discuss more than a thousand theorems on plane geometry and hopefully it can help to the students preparing for the Olympiad competition.

Please share this to the people may need it.

Thank you!
search
geometry
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Beautiful geo but i cant solve this

by phonghatemath, Jun 1, 2025, 4:48 PM

Given triangle $ABC$ inscribed in $(O)$. Two points $D, E$ lie on $BC$ such that $AD, AE$ are isogonal in $\widehat{BAC}$. $M$ is the midpoint of $AE$. $K$ lies on $DM$ such that $OK \bot AE$. $AD$ intersects $(O)$ at $P$. Prove that the line through $K$ parallel to $OP$ passes through the Euler center of triangle $ABC$.

Sorry for my English!
This post has been edited 1 time. Last edited by phonghatemath, Yesterday at 4:52 PM
Reason: i had a mistake while translating
geometry
angle bisector
Euler
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Projective geo

by drmzjoseph, Mar 6, 2025, 4:26 PM

Any pure projective solution? I mean no metrics, Menelaus, Ceva, bary, etc
Only pappus, desargues, dit, etc
Btw prove that $X',P,K$ are collinear, and $P,Q$ are arbitrary points
Attachments:
geometry
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How many cases did you check?

by avisioner, Jul 17, 2024, 12:01 PM

Determine all ordered pairs $(a,p)$ of positive integers, with $p$ prime, such that $p^a+a^4$ is a perfect square.

Proposed by Tahjib Hossain Khan, Bangladesh
This post has been edited 1 time. Last edited by avisioner, Jul 20, 2024, 4:57 PM
Reason: Proposer name added
number theory
IMO Shortlist
AZE IMO TST
TST
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Symmetric integer FE

by a_507_bc, Jul 1, 2023, 12:36 PM

Find all functions $f: \mathbb{Z} \to \mathbb{Z}$, such that $$f(x+y)((f(x) - f(y))^2+f(xy))=f(x^3)+f(y^3)$$for all integers $x, y$.
This post has been edited 1 time. Last edited by a_507_bc, Jul 1, 2023, 12:36 PM
algebra
functional equation
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Conditional geo with centroid

by a_507_bc, Jul 1, 2023, 12:28 PM

In a scalene triangle $ABC$ with centroid $G$ and circumcircle $\omega$ centred at $O$, the extension of $AG$ meets $\omega$ at $M$; lines $AB$ and $CM$ intersect at $P$; and lines $AC$ and $BM$ intersect at $Q$. Suppose the circumcentre $S$ of the triangle $APQ$ lies on $\omega$ and $A, O, S$ are collinear. Prove that $\angle AGO = 90^{o}$.
geometry
homothety
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HMMT Thursday (2/16)

by ChickenAgent2227-_-, Feb 26, 2023, 8:28 AM

HMMT was incredibly fun and rewarding and, to be honest, both as intense as I could handle and too little for me to be satisfied. I'll be posting by days periodically, as I usually do, because there’s enough content to cover multiple posts and I gotta beat the blogrolllll.

Thursday (2/16)

We got to the airport on Friday night with the two San Diego teams: AJ, BG, DaL, DeL, EQ, EL, ES, EL, JR, JL, JZ, ML, NM, RM, RS. Once we got to our gate, three friends of a friend who aren’t math sweats, SB, MC, AW, came to find me because it turns out that they were at the same airport as well on a Boston visit! They found me in an unfortunately recognizable condition though, although this requires context…

See, HMMT has been my only opportunity to reunite with some friends from MOP who I’ve missed very much. I’m terrible at maintaining contact through social media because texting is hard and annoying, so an in-person meeting is incredibly precious and I have been looking forward to this for a good six months since June. But for such a monumental reunion, one must duly prepare monumentally!

In the week before, I had prepared myself a disguise: I brought two wigs, a beanie, layers beyond layers of jackets to change my physical shape. I had sunglasses so cheap they broke if you touched them too hard. I borrowed a mask, I recorded a clip of a friend speaking so I could imitate his voice, because I judged his voice to be sufficiently different from mine. I had planned to get make-up, although that fell through at the last moment. I was going to put pebbles in my shoes so they wouldn’t catch me by my characteristic gait. I was going to troll them so hard and end by stealing IZ’s water bottle. I played the scene out in my head a million different times, daydreamed a million laughs and screams.

So I was at the airport trying on my wigs, and that’s how SB, MC, AW saw me when they pulled up. Everyone was like, "DX WHAT ARE YOU DOING" and I only had to explain repeatedly “Who’s DX? I’m Jeremy!”


Anyways, I sat next to BG on the flight. The ceiling was tiled like the AoPS logo. A weird humming sound appeared, so BG who is apparently an expert at planes because he rode one a few years ago insisted “this is definitely not a normal sound this is bad” and I laughed at him.

DeL had warned us ahead of time not to do any math right before HMMT, so we did a mock USAMO together, solving #1,2,4,5 on a 6-hour flight. I also spent some time practicing the voice of my alias. I got too sleepy near the end, so I leaned my head on the TV screen to catch some Z’s while BG continued solving math problems (he ended up solving #6).

Except every time the plane bounced up and down, my head bounced up and down on the screen and hit random buttons, and on not one, but two separate occasions, BG woke me up to ask why there was an adult scene playing on my TV screen. I DON’T KNOW BG! Why don’t you tell me why the TV screen next to you always goes to adult scenes while I’m unconscious?
This post has been edited 2 times. Last edited by ChickenAgent2227-_-, Feb 26, 2023, 10:34 PM
HMMT

People live in Kansas?

by jj_ca888, Aug 28, 2020, 6:17 PM

In triangle $\triangle ABC$, let $E$ and $F$ be points on sides $AC$ and $AB$, respectively, such that $BFEC$ is cyclic. Let lines $BE$ and $CF$ intersect at point $P$, and $M$ and $N$ be the midpoints of $\overline{BF}$ and $\overline{CE}$, respectively. If $U$ is the foot of the perpendicular from $P$ to $BC$, and the circumcircles of triangles $\triangle BMU$ and $\triangle CNU$ intersect at second point $V$ different from $U$, prove that $A, P,$ and $V$ are collinear.

Proposed by Andrew Wen and William Yue
This post has been edited 3 times. Last edited by jj_ca888, Aug 28, 2020, 9:09 PM
S(J)MO
geometry
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2019 Iberoamerican Mathematical Olympiad, P1

by jbaca, Sep 15, 2019, 6:58 PM

For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
number theory
Digits
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Product of Sum

by shobber, Aug 9, 2006, 4:41 AM

Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]
inequalities
inequalities unsolved
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All who attend MOP are transformed by it. Character, perspective–math is the least of it. Here is my story. Let this blog become folklore. Let its name rest in your minds. And when you make MOP, and you want to look ahead–remember me.

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ChickenAgent2227-_-
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