Impossible to search, classic graph problem

by AshAuktober, Mar 30, 2025, 5:19 PM

Prove that any graph $G=(V,E)$ with $|V|=|E|-1$ has at least two cycles in it.
combinatorics
graph theory
L

Easy problem

by Hip1zzzil, Mar 30, 2025, 1:18 PM

$(C,M,S)$ is a pair of real numbers such that

$2C+M+S-2C^{2}-2CM-2MS-2SC=0$
$C+2M+S-3M^{2}-3CM-3MS-3SC=0$
$C+M+2S-4S^{2}-4CM-4MS-4SC=0$

Find $2C+3M+4S$.
This post has been edited 1 time. Last edited by Hip1zzzil, 5 hours ago
Reason: Rr
algebra
L

When is this well known sequence periodic?

by Assassino9931, Mar 30, 2025, 1:17 PM

Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.
algebra
periodic
L

Polynomials and their shift with all real roots and in common

by Assassino9931, Mar 30, 2025, 1:12 PM

We call two non-constant polynomials friendly if each of them has only real roots, and every root of one polynomial is also a root of the other. For two friendly polynomials \( P(x), Q(x) \) and a constant \( C \in \mathbb{R}, C \neq 0 \), it is given that \( P(x) + C \) and \( Q(x) + C \) are also friendly polynomials. Prove that \( P(x) \equiv Q(x) \).
This post has been edited 1 time. Last edited by Assassino9931, 5 hours ago
algebra
polynomial
L

Train yourself on folklore NT FE ideas

by Assassino9931, Mar 30, 2025, 12:39 PM

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.
This post has been edited 1 time. Last edited by Assassino9931, 5 hours ago
functional equation
number theory
Divisibility
primes
Sophie Germain identity
function
L

Geo challenge on finding simple ways to solve it

by Assassino9931, Mar 30, 2025, 12:35 PM

Let $ABC$ be an acute scalene triangle inscribed in a circle \( \Gamma \). The angle bisector of \( \angle BAC \) intersects \( BC \) at \( L \) and \( \Gamma \) at \( S \). The point \( M \) is the midpoint of \( AL \). Let \( AD \) be the altitude in \( \triangle ABC \), and the circumcircle of \( \triangle DSL \) intersects \( \Gamma \) again at \( P \). Let \( N \) be the midpoint of \( BC \), and let \( K \) be the reflection of \( D \) with respect to \( N \). Prove that the triangles \( \triangle MPS \) and \( \triangle ADK \) are similar.
This post has been edited 1 time. Last edited by Assassino9931, 5 hours ago
geometry
Angle Chasing
similarity
circumcircle
L

VERY HARD MATH PROBLEM!

by slimshadyyy.3.60, Mar 29, 2025, 10:49 PM

Let a ≥b ≥c ≥0 be real numbers such that a^2 +b^2 +c^2 +abc = 4. Prove that
a+b+c+(√a−√c)^2 ≥3.
L

Problem 2

by Functional_equation, Jun 6, 2020, 9:16 AM

$a,b,c$ are positive integer.
Solve the equation:
$ 2^{a!}+2^{b!}=c^3 $
This post has been edited 1 time. Last edited by Functional_equation, Jun 9, 2020, 6:15 AM
number theory
L

Functional equation

by Dadgarnia, Apr 15, 2018, 12:15 PM

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions:
a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$
b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval.

Proposed by Navid Safaei
This post has been edited 1 time. Last edited by Dadgarnia, Apr 15, 2018, 12:15 PM
function
algebra
functional equation
L

Intersection of a cevian with the incircle

by djb86, May 27, 2012, 9:23 AM

The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.
geometry
analytic geometry
power of a point
geometry unsolved
L

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Submit

  • done!
    $~~~~$

    by pupitrethebean, Mar 19, 2025, 2:05 PM

  • Hi! Contrib?

    by RandomeMids, Mar 13, 2025, 2:22 PM

  • aww tysmmmm

    by pupitrethebean, Feb 3, 2025, 10:19 PM

  • revive the shoutbox lol

    i like your blog : )

    by witchofblackbirdpond, Feb 3, 2025, 7:56 PM

  • revive the shoutbox

    by pupitrethebean, Jan 20, 2025, 4:42 AM

  • d
    e
    d
    .$~~$

    by ujulee, Nov 26, 2024, 11:37 PM

  • bruhhh nahhhh

    by pupitrethebean, Oct 13, 2024, 3:22 PM

  • this blog is so aesthetic it's crazy

    by Technodoggo, Oct 12, 2024, 11:16 PM

  • ehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh soon :>

    by pupitrethebean, Oct 12, 2024, 1:29 AM

  • pupitre you should totally promote meee

    by CC_chloe, Oct 11, 2024, 7:34 PM

  • admins and contribs r different colors

    by pupitrethebean, Oct 8, 2024, 3:01 PM

  • why are certain usernames different colors :thonk:

    by Technodoggo, Oct 8, 2024, 12:44 AM

  • spoooooky

    by blair_givenchy, Oct 3, 2024, 12:24 PM

  • okie dokes

    by pupitrethebean, Oct 3, 2024, 1:55 AM

  • could i pls contrib as well? and if it's alright could i have display name of "Lydia"?

    by Hestu_the_Bestu, Oct 2, 2024, 5:52 PM

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y are there 24 posts in 14 day
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About Owner
  • Posts: 4
  • Joined: Jan 19, 2023
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  • Blog created: Mar 31, 2023
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