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such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that for all 
,
is a prime number, 
are natural numbers (
). It is known that 
. Prove that 
.
such that 
and
.
and a plane 
, parallel with the edges 
and 
.
.
and 
,
, let 
denote the increasing arithmetic sequence of integers whose first term is 
and whose common difference is . For example, 
is the sequence 
. For how many values of does contain the term 
?
is the set of positive real solutions to the system![\[
x^3 + y^3 + z^3 = x + y + z
\]](http://latex.artofproblemsolving.com/7/7/a/77a748a8b55a6b44761bced9bb1031193b6127dc.png)
![\[
x^2 + y^2 + z^2 = xyz
\]](http://latex.artofproblemsolving.com/9/3/a/93a4dffc6fd17b9f19f738d3b3865f56aa0151a9.png)
then find the number of elements in .
satisfy![\[
a + \frac{1}{a + 2015} = b - 4030 + \frac{1}{b - 2015}
\]](http://latex.artofproblemsolving.com/8/c/0/8c0b0cfdd569ea250e97590a3364794920d86e04.png)
and 
. Determine the value of![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
and 
are same then obviously your answer is wrong. Actually the answer would be 
I guess. Trying to prove.....
case is trivial which gives us 
....and the rest as follows
doesn't necessarily divide that part....
be a prime such that 
for 
. Using![\[
a^3+b^3+1 - 3ab = (a+b+1)(a^2+b^2+1-ab-a-b)
\]](http://latex.artofproblemsolving.com/e/9/7/e978baa1bbf867ec90243096ff0c1edf3725f438.png)
we get 
. Suppose first that 
(the case 
is symmetric). Then, 
forces 
which together with 
forces 
. So, 
, where 
are arbitrary are both solution families.
. Then 
and 
. Since 
, we must have 
, giving the family 
where 
are arbitrary.
and 
Prove that![$$ a+b+kc^3\geq\sqrt[4]{\frac{4k} {27}}$$](http://latex.artofproblemsolving.com/5/6/1/561a990e5885863c77d454ffa3f1973e1bbf60b4.png)
![$$ a+b+kc^4\geq\frac{5} {8}\sqrt[5]{\frac{k} {2}}$$](http://latex.artofproblemsolving.com/4/b/1/4b14911672688250c909571d788aaf10724462eb.png)
Where 
and 
Prove that
numebr on his list? Write its last two digits as your answer.
grid of square tiles. Robert chooses four of these squares uniformly at random. The probability that the centers of these four squares form a square themselves is 
where 
are coprime naturals. Write the largest 
digit odd factor of 
.
.
which touches one face of the cube and passes through all 
vertices of the face opposite to it. Find the integer closest to 
are distinct numbers in AP such that 
are also in AP. Find the smallest possible value of 
(in degrees).
have a greatest common divisor 
.
,
,
,
,
.
divides 
and 
,
,
for all 
.
for some non-negative integers 
.
is odd then one of this numbers has to be even and one odd and then this pair works in this problem 
)
.
.
and