[/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, and 1 other user
The Ababi alphabet consists of letters A and B, and the words in the Ababi language are precisely those that can be formed by the following two rules:
1) A is a word.
2) If s is a word, then
and 
are words, where 
denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and 
denotes the concatenation of x and y.
The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules:
1) A is a word.
2) If s is a word, and are words, where is defined as above and is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x.
Prove that the two languages consist of the same words.
1) A is a word.
2) If s is a word, then
and 
are words, where 
denotes a word that is obtained by replacing all letters A in s with letters B, and vice versa; and 
denotes the concatenation of x and y.The Ululu alphabet consists also of letters A and B and the words in the Ululu language are precisely those that can be formed by the following two rules:
1) A is a word.
2) If s is a word,
and are words, where is defined as above and is a word obtained from words x and y of equal length by writing the letters of x and y alternatingly, starting from the first letter of x.Prove that the two languages consist of the same words.
and having length 
.
this is clearly the case. Then for words of length 
consider their first character (A) and the remaining block of 
characters (which by assumption ranges through all possible strings of As and Bs); then 
can range over all strings of length 
.
,
;
beginning with 
this is clearly the case.
words of length 
you seem to be imagining 
.)
for the second operation (and if the original problem uses the same notation for two different operations, that's really bad form). For words of length 1, it's trivial. Take a word 
,
for some 
equal to either 
or 
.
be equal to either 
.
,
.
