thanks u!
by Ruji2018252, Apr 3, 2025, 5:56 PM
find all 
and
![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](//latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
Fneqn or Realpoly?
by Mathandski, Apr 3, 2025, 5:46 PM
Find all polynomials 
with real coefficients obeying
![\[P(x) P(x+1) = P(x^2 + x + 1)\]](//latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.high school maths
by aothatday, Apr 3, 2025, 2:27 PM
find 
such that:
such that:
This post has been edited 2 times. Last edited by aothatday, Today at 2:30 PM
Coaxial circles related to Gergon point
by Headhunter, Apr 3, 2025, 2:48 AM
Hi, everyone.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.
Let the circumcircles of
, 
, 
be 
, 
, 
respectively.
Reflect in 
and then we get the circle 
Reflect in 
and then the circle 
Reflect in 
and then the circle 
Prove that , , are coaxial.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.Let the circumcircles of
, 
, 
be 
, 
, 
respectively.Reflect
in 
and then we get the circle 
Reflect
in 
and then the circle 
Reflect
in 
and then the circle 
Prove that
, , are coaxial.
such that 
D1019 : Dominoes 2*1
by Dattier, Mar 26, 2025, 8:18 AM
I have a 9*9 grid like this one:
We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:
Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:
We choose 5 white squares on the lower triangle, 5 black squares on the upper triangle and one on the diagonal, which we remove from the grid.
Like for example here:
Can we completely cover the grid remove from these 11 squares with 2*1 dominoes like this one:
D1018 : Can you do that ?
by Dattier, Mar 24, 2025, 6:01 AM
We can find 
, such that 
and 
.
For example :
Can you find such that 
is prime, 
with 
and ?
, such that 
and 
.For example :
Can you find
such that 
is prime, 
with 
and ?D1010 : How it is possible ?
by Dattier, Mar 10, 2025, 10:49 AM
Is it true that
?
A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975
B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
?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
Something nice
by KhuongTrang, Nov 1, 2023, 12:56 PM
Problem. Given 
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that 
Prove that
This post has been edited 2 times. Last edited by KhuongTrang, Nov 19, 2023, 11:59 PM
iran tst 2018 geometry
by Etemadi, Apr 17, 2018, 3:34 PM
Let 
be the circumcircle of isosceles triangle (
). Points and 
lie on and respectively such that 
.
and intersect at 
. Prove that the tangents from 
and 
to the incircle of 
(different from ) are concurrent on .
Proposed by Ali Zamani, Hooman Fattahi
be the circumcircle of isosceles triangle (
). Points and 
lie on and respectively such that 
.
and intersect at 
. Prove that the tangents from 
and 
to the incircle of 
(different from ) are concurrent on .Proposed by Ali Zamani, Hooman Fattahi
This post has been edited 6 times. Last edited by Etemadi, Apr 21, 2018, 3:43 PM
1. Algebra and sigma algebra on a set
by adityaguharoy, Feb 28, 2018, 3:28 PM
be an arbitrary set. Then a collection 
for each 
the set 
for every finite sequence 
of elements each 
the union 
of elements each 
,Let
be an infinite set. And let 
be the collection of all subsets 
of such that either or 
is finite. Prove that is an algebra on , but not a sigma algebra on .
then 
to 
be such a function.
)
be the union 
then 
is an infinite set. Thus, A related exercise :
Let
be an . Let be the collection of all subsets of such that either is countable or 
is countable.Is
a 
algebra on ?This post has been edited 2 times. Last edited by adityaguharoy, Feb 28, 2018, 4:16 PM
The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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