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and 
and 
are real numbers for which the following inequality is satisfied :![\[\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\]](http://latex.artofproblemsolving.com/5/e/4/5e4b3bdc2d5b71e42ba3e6339845794625080eb5.png)
. Prove that 
.
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
and 
be positive real numbers such that 
. Prove that
When equality holds?
be an isosceles triangle such that 
. Let 
be a dot on the 
side.
at point intersects the circumcircle of 
at 
. Prove that CD
AB
, find the number of terms with coefficient divisible by 
.
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.
,
.
Prove that
and 
.
Let 
.
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
.
,
,
,
,
.
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