Valuable subsets of segments in [1;n]

by NO_SQUARES, Apr 3, 2025, 8:34 PM

The integer $n \geqslant 2$ is given. Let $A$ be set of all $n(n-1)/2$ segments of real line of type $[i, j]$, where $i$ and $j$ are integers, $1\leqslant i<j\leqslant n$. A subset $B \subset A$ is said to be valuable if the intersection of any two segments from $B$ is either empty, or is a segment of nonzero length belonging to $B$. Find the number of valuable subsets of set $A$.
combinatorics
L

thanks u!

by Ruji2018252, Apr 3, 2025, 5:56 PM

find all $f: \mathbb{R}\to \mathbb{R}$ and
\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]
functional equation
IMO
L

Fneqn or Realpoly?

by Mathandski, Apr 3, 2025, 5:46 PM

Find all polynomials $P$ with real coefficients obeying
\[P(x) P(x+1) = P(x^2 + x + 1)\]for all real numbers $x$.
algebra
polynomial
functional equation
L

n=y^2+108

by Havu, Apr 3, 2025, 4:30 PM

Given the positive integer $n = y^2 + 108$ where $y \in \mathbb{N}$.
Prove that $n$ cannot be a perfect cube of a positive integer.
number theory
L

high school maths

by aothatday, Apr 3, 2025, 2:27 PM

find $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:
$(x-y)(f(x)+f(y)) \leq f(x^2-y^2)$
This post has been edited 2 times. Last edited by aothatday, Today at 2:30 PM
help me pls
inequalities
L

Functional equations

by hanzo.ei, Mar 29, 2025, 4:33 PM

Find all $f: \mathbb R_+ \rightarrow \mathbb R_+$ such that $f(xf(y)+f(x))=yf(x+yf(x)) \: \forall \: x,y \in \mathbb R_+$
algebra
functional equation
L

Unsolved NT, 3rd time posting

by GreekIdiot, Mar 26, 2025, 11:40 AM

Solve $5^x-2^y=z^3$ where $x,y,z \in \mathbb Z$
Hint
This post has been edited 2 times. Last edited by GreekIdiot, Mar 26, 2025, 11:41 AM
number theory
modular arithmetic
Exponents
unsolved
L

D1018 : Can you do that ?

by Dattier, Mar 24, 2025, 6:01 AM

We can find $A,B,C$, such that $\gcd(A,B)=\gcd(C,A)=\gcd(A,2)=1$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$.

For example :

$C=20$
$A=47650065401584409637777147310342834508082136874940478469495402430677786194142956609253842997905945723173497630499054266092849839$

$B=238877301561986449355077953728734922992395532218802882582141073061059783672634737309722816649187007910722185635031285098751698$

Can you find $A,B,C$ such that $A>3$ is prime, $C,B \in (\mathbb Z/A\mathbb Z)^*$ with $o(C)=(A-1)/2$ and $$\forall n \in \mathbb N^*, (C^n \times B \mod A) \mod 2=0 $$?
arithmetic
number theory
L

D1010 : How it is possible ?

by Dattier, Mar 10, 2025, 10:49 AM

Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
This post has been edited 6 times. Last edited by Dattier, Mar 16, 2025, 10:10 AM
number theory
Computer solutions
L

iran tst 2018 geometry

by Etemadi, Apr 17, 2018, 3:34 PM

Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.

Proposed by Ali Zamani, Hooman Fattahi
This post has been edited 6 times. Last edited by Etemadi, Apr 21, 2018, 3:43 PM
geometry
L

"Where wisdom and valor fail, all that remains is faith. . . And it can overcome all." -Toa Mata Tahu

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  • The most viewed blog of all time

    by giangtruong13, Feb 24, 2025, 1:37 PM

  • wow this blog is ancient
    it started when I was an infant
    wow and like 1.5 million views

    by nmlikesmath, Oct 13, 2024, 10:09 PM

  • no ur cool

    tbh it scares me too I still dont know where those views came from

    by EpicSkills32, Oct 5, 2024, 9:44 AM

  • wowwww this blog is so cool

    by QueenArwen, Oct 4, 2024, 8:25 AM

  • it scares me how many views this blog has

    by prettyflower, Sep 29, 2024, 9:38 AM

  • bruh
    this blog started when i was like 2 months old

    by TimmyL, Sep 27, 2024, 9:18 AM

  • 1579363 visits rn, curious how many views (if any) this blog gets now

    by EpicSkills32, Sep 16, 2024, 10:47 AM

  • interesting lol

    by EpicSkills32, Jul 27, 2024, 11:27 AM

  • EpicSkills32 reaped on Jul 27, 18:44:19 and gained 10 minutes, 57 seconds

    orz orz orz

    highest raw of the game so far

    by DuoDuoling0, Jul 27, 2024, 10:47 AM

  • whoa this blog is old

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  • greetings

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  • hi$ $

    by URcurious2, Apr 13, 2024, 4:10 AM

  • orzorzorz can i have contrib :)

    by dbnl, Aug 11, 2023, 10:18 PM

  • 1.5M views pro

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