[/quote]
,
and 
.
.
of degree 
.![$[a;b]$](http://latex.artofproblemsolving.com/0/0/6/0066f4b2714f88cdaf883b00f877aa1355b7f9ef.png)
and Maxim will give in response the maximum value of 
![\[
v_p(n!) \leq \frac{n}{p - 1}
\]](http://latex.artofproblemsolving.com/5/7/c/57ca4d0a6532fb4b97687a97b9c1fb35ad02b9f2.png)
and 
Prove that
Let 
and 
Prove that
not isosceles, the incircle 
.
is the intersection of 
.
is the projection of 
onto 
.
cut 
.
is the projection of 
onto 
.
cut 
respectively. Prove 
are collinear
be the set of non-negative integers and 
be the set of positive real numbers. Let 
be a function such that 
and 
for all integers 
,
for all integers 
.
.
which satisfy 
![$ \frac{a+b+c}{3} \ge \sqrt[10]{\frac{a^3+b^3+c^3}{3}}$](http://latex.artofproblemsolving.com/c/d/a/cdae05ad34239366b0dd3921243522c711b6ae58.png)
method, I can assum 
![$ \frac{2x+\frac{1}{x^2}}{3} \ge \sqrt[10]{ \frac{2x^3+ \frac{1}{x^6}}{3}} $](http://latex.artofproblemsolving.com/c/d/4/cd481672bee1e87d8c97ac623436e0ec75cfa73a.png)
are given such that 
.
in natural numbers m and n.
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
intersects 
at 
.
cuts segment 
at 
respectively. If 
and 
,
equals 
.
are points on a circle 
,
,
.
at 
is on 
such that 
.
intersects 
.
is the reflection of 
.
lies on the same circle.
Y by Adventure10, Mango247
Two players A and B play the following game. Initially, there are 
equal positive integers 
written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with 
; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?
equal positive integers 
written on a blackboard. A begins and the players move alternately. The player to move chooses one of the non-zero numbers on the board. If this number k is the smallest among all positive integers on the board, the player replaces it with 
; if not, the player replaces it with the smallest positive number on the board. The player who first turns all the numbers into zeroes, wins. Who wins if both players use their best strategies?
.
,
,
.
