[/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)
.
,
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 
?
satisfying
![\[
\text{Find all non-negative integers } (a, b, c, n) \text{ such that}
\]](http://latex.artofproblemsolving.com/9/5/9/9591f7c842d1b6afcba11b15b3ea3f9e01bc937a.png)
![\[
2^a + 3^b + 5^c = n!
\]](http://latex.artofproblemsolving.com/b/c/6/bc6b0754dd874f646fcab124c91583b620db273c.png)
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
Let 
be a finite abelian group. There is a magic box 
. At any point, an element of may be added to the box and all elements belonging to the subgroup (of ) generated by the elements currently inside are moved from outside to inside (unless they are already inside). Initially 
contains only the group identity, 
. Alice and Bob take turns moving an element from outside to inside it. Alice moves first. Whoever cannot make a move loses. Find all for which Bob has a winning strategy.
be a finite abelian group. There is a magic box 
. At any point, an element of may be added to the box and all elements belonging to the subgroup (of ) generated by the elements currently inside are moved from outside to inside (unless they are already inside). Initially 
contains only the group identity, 
. Alice and Bob take turns moving an element from outside to inside it. Alice moves first. Whoever cannot make a move loses. Find all for which Bob has a winning strategy.
