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 
)
.
.
such that 
and
.
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).
and a plane 
,
and 
.
.
and 
,
,
denote the increasing arithmetic sequence of integers whose first term is 
and whose common difference is 
is the sequence 
.
?
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 
.![\[
\frac{ab}{2015} - a + b.
\]](http://latex.artofproblemsolving.com/1/f/6/1f620dea7a0c769137bf13004916e8b283e00eee.png)
Y by ShahinH, lafius, ahaanomegas, fireonice18, StudentIV, AdithyaBhaskar, m1234567, Taekyukim02, Severus, Uros, mathleticguyyy, fjm30, hansduran0123, Adventure10, and 14 other users
See the attachment. 
This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file.
This is a nice collection of trigonometry problems. I searched but didn't find a nice collection about trigonometry, so searched trigonometry problems posted on AoPS and made this file.
This post has been edited 1 time. Last edited by Amir Hossein, Nov 21, 2017, 9:49 AM
Reason: Added the TeX file.
Reason: Added the TeX file.
!
.
.
.
,
and 
.
.
rearranges 
.
now we have to find the value of
by some manipulation we have
now we use 
so we deduce
