[/quote]
,
![\[
f(x) = \sqrt{1 - x^a}, \quad a \geq 0
\]](http://latex.artofproblemsolving.com/e/f/1/ef14689f1894683256f9a0155037788366496929.png)
![\( x \in [0,1] \)](http://latex.artofproblemsolving.com/7/6/d/76d950c78d04a107e9fcb49b26ab353531c93b92.png)
.
,
be positive integers.Define injective function 
from 
to itself such that 
.
divides 
.
board that can be colored so that for each square in the board, at most one square adjacent to it is colored.
such that 
.
that divides 
for the positive integer 
.
points are collinear in Euclidean plane, such that Shining cannot find a group of three points such that their circumcircle contains none of the other remaining points. Is he always able to do so?
of positive integers such that 
in all of this cases :
3-Every prime number less than 
can be represented in the form 
by taking an Appropriate pair 
and 
. The number 
is the least prime number that cannot be represented as a sum or an difference of a power of 
, 
, 
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
be real positive numbers such that: 
.
,
such that 
is a perfect square? (Here 
denotes the integer part of the real number 
?
is a power series for which each coefficient 
is 
or 
.
,
must be irrational.
.
s.t. 
.
,![$K=\mathbb{Q}[\cos(\dfrac{2}{3}\arctan(3\sqrt{3}))]$](http://latex.artofproblemsolving.com/a/2/e/a2efd64f27687f9ba5239cc4e0ba49d469d1c8c9.png)
.
.
and not its conjugate transpose.
.
;
Y by Creativename27
How many spheres are needed to shield a point source of light?
Unfortunately, I didn't find a suitable solution for it on the page below:
https://artofproblemsolving.com/community/c4h1469498p8521602
Here too, two different solutions are given:
https://math.stackexchange.com/questions/2791186
Unfortunately, I didn't find a suitable solution for it on the page below:
https://artofproblemsolving.com/community/c4h1469498p8521602
Here too, two different solutions are given:
https://math.stackexchange.com/questions/2791186
This post has been edited 3 times. Last edited by mofidy, Mar 11, 2025, 1:08 PM
spheres suffice. Indeed, let 
be the light source, 
be a regular tetrahedron inscribed in the unit sphere 
with center 
with center 
passing through a point very close to 
.
shield 
.
are coplanar, then the line through 
does not cut any sphere. Otherwise define 
and 
.
does not cut any sphere since 
makes acute and equal angles with 
.
