AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
probability of coming up heads and 
probability of coming up tails. The expected number of flips necessary to first see the sequence 
in that consecutive order can be written as 
for relatively prime positive integers 
, 
. Find the number of factors of .
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
.
, 
, 
be positive real numbers such that 
. Prove that ![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
and be positive real numbers satisfying 
and 
Prove that![\[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\]](http://latex.artofproblemsolving.com/0/a/b/0ab4b4e6c28a1ae67788525642661dbcd04d0bd8.png)
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
such for any :
.
satisfy,
Where {
} is a permutation of . Prove that 
be an integer. We are given a balance and weights of weight 
. We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
be a right triangle of area 
m
. Consider the triangle 
such that 
is the symmetric of X wrt side 
, 
is the symmetric of 
wrt side 
and 
is the symmetric of 
wrt side 
. Calculate the area of the triangle .
be a right triangle of area m . Consider the triangle such that is the symmetric of X wrt side , is the symmetric of wrt side and is the symmetric of wrt side . Calculate the area of the triangle .
,
equals 1.
.
equals 3.