Inspired by SunnyEvan

by sqing, May 17, 2025, 2:59 PM

Let $ a,b,c $ be reals such that $ a^2+b^2+c^2=2(ab+bc+ca). $ Prove that$$ \frac{1}{12} \leq \frac{a^2b+b^2c+c^2a}{(a+b+c)^3} \leq \frac{5}{36} $$Let $ a,b,c $ be reals such that $ a^2+b^2+c^2=5(ab+bc+ca). $ Prove that$$ -\frac{1}{25} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{197}{675} $$
This post has been edited 3 times. Last edited by sqing, a few seconds ago
inequalities proposed
algebra
L

A circle tangent to the circumcircle, excircles related

by kosmonauten3114, May 17, 2025, 1:12 PM

Let $ABC$ be a scalene triangle with excircles $\odot(I_A)$, $\odot(I_B)$, $\odot(I_C)$. Let $\odot(A')$ be the circle which touches $\odot(I_B)$ and $\odot(I_C)$ and passes through $A$, and whose center $A'$ lies outside of the excentral triangle of $\triangle{ABC}$. Define $\odot(B')$ and $\odot(C')$ cyclically. Let $\odot(O')$ be the circle externally tangent to $\odot(A')$, $\odot(B')$, $\odot(C')$.

Prove that $\odot(O')$ is tangent to the circumcircle of $\triangle{ABC}$ at the anticomplement of the Feuerbach point of $\triangle{ABC}$.
Attachments:
geometry
Triangle
circumcircle
excircles
tangent circles
Feuerbach Point
L

3 var inequality

by SunnyEvan, May 17, 2025, 1:05 PM

Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{53}{2}-9\sqrt{14} \leq \frac{8(a^3b+b^3c+c^3a)}{27(a^2+b^2+c^2)^2} \leq \frac{53}{2}+9\sqrt{14} $$
This post has been edited 2 times. Last edited by SunnyEvan, 2 hours ago
inequalities
L

Insspired by Shandong 2025

by sqing, May 17, 2025, 9:23 AM

Let $ a,b,c>0,abc>1$. Prove that$$ \frac {abc(a+b+c+ab+bc+ca+3)}{ abc-1}\geq \frac {81}{4}$$$$ \frac {abc(a+b+c+ab+bc+ca+abc+2)}{ abc-1}\geq 12+8\sqrt{2}$$
This post has been edited 1 time. Last edited by sqing, 6 hours ago
inequalities
inequalities proposed
algebra
L

Find the minimum

by sqing, May 17, 2025, 9:12 AM

Let $ a,b,c>0,abc>1$. Find the minimum value of $ \frac {abc(a+b+c+8)}{abc-1}. $
inequalities proposed
algebra
L

A point on BC

by jayme, May 17, 2025, 6:08 AM

Dear Mathlinkers,

1. ABC a triangle
2. 0 the circumcircle
3. D the pole of BC wrt 0
4. B', C' the symmetrics of B, C wrt AC, AB
5. 1b, 1c the circumcircles of the triangles BB'D, CC'D
6. T the second point of intersection of the tangent to 1c at D with 1b.

Prove : B, C and T are collinear.

Sincerely
Jean-Louis
geometry
L

Prove that the triangle is isosceles.

by TUAN2k8, May 16, 2025, 9:55 AM

Given acute triangle $ABC$ with two altitudes $CF$ and $BE$.Let $D$ be the point on the line $CF$ such that $DB \perp BC$.The lines $AD$ and $EF$ intersect at point $X$, and $Y$ is the point on segment $BX$ such that $CY \perp BY$.Suppose that $CF$ bisects $BE$.Prove that triangle $ACY$ is isosceles.
geometry
Triangle
L

Bound of number of connected components

by a_507_bc, Aug 12, 2023, 12:44 PM

Let $G$ be a connected graph and let $X, Y$ be two disjoint subsets of its vertices, such that there are no edges between them. Given that $G/X$ has $m$ connected components and $G/Y$ has $n$ connected components, what is the minimal number of connected components of the graph $G/(X \cup Y)$?
This post has been edited 3 times. Last edited by a_507_bc, Aug 13, 2023, 7:00 AM
combinatorics
L

Israel Number Theory

by mathisreaI, Jul 13, 2022, 2:55 AM

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]
This post has been edited 2 times. Last edited by v_Enhance, Jul 13, 2023, 6:28 PM
Reason: missing full stop
number theory
Divisibility
LTE Lemma
IMO
IMO 2022
Diophantine equation
L

Bounds on degree of polynomials

by Phorphyrion, Jun 11, 2022, 11:34 AM

For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds:

For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, all of the same size, let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which
\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]
a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$.

b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.
number theory
Polynomials
olympic revenge
algebra
polynomial
L

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  • omg tysmmmm

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  • haiiiiiii cute blog <3

    by yaxuan, May 8, 2025, 8:29 AM

  • done!
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    by pupitrethebean, Mar 19, 2025, 2:05 PM

  • Hi! Contrib?

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

  • aww tysmmmm

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  • revive the shoutbox lol

    i like your blog : )

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

  • revive the shoutbox

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

  • bruhhh nahhhh

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  • this blog is so aesthetic it's crazy

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  • ehhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh soon :>

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

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  • admins and contribs r different colors

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my brother
Name
Netflix
nice
no ac
no comprendo
no school
Nooooo
Oh
oh no
oh no children
Old
oops
ootd
orange
panera
Photos
picture
pizza
Power
project
qotd
questionable
Rabbit
rating
remember kids
reminder
RIP
ruff ruff
sadness
sajfdjasf
seansuksd
send help
Sick
skip
smort
so random
sobbing
sobs
song
SOUP
Spanish
springbreak
stairs
Stinky
story
stuff
taipei
taipei 101
teasuckspp
this is a bad idea
Thoughts
Tired
Travel
twoset
uh oh
uju dont say smooth hand
update
Vacation
vans
vansisstinky
Warm
wedding
wevideo
what
what an L
whenitsnotday
work
WWI
y are there 24 posts in 14 day
ye
YEEE
About Owner
  • Posts: 4
  • Joined: Jan 19, 2023
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  • Blog created: Mar 31, 2023
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