AoPS Academy
and 
(unordered i.e. choosing 
is same as choosing 
from the set 
such that difference between them is atmost 10, i.e. 
, such that 
, 
and 
. which are not perfect squares. Calculate the 
-th term of the sequence.
, let 
be the midpoint of side 
and 
be the midpoint of segment 
. Lines 
and 
intersect at 
. Show that 
.
such that 
is a square perfect.
friends in this. However, if two people are not friends at least they have a friend in this party. What is the largest possible number of people in the party?
of positive integers which satisfy the equation 
be a triangle with side lengths 
, 
, 
, incenter 
and circumcircle 
, and let 
be the circle that is tangent to 
, , at ,, , respectively. If 
,
, 
denote the incenters of 
, 
, 
, respectively, find the measure, in degrees, of the largest angle in 
.
. Multiplying by 
on both sides and resubstituting 
, we find 
. Continuing this process, we find 
, 
, and so forth. We claim that 
We already have our base case. 
. Multiplying by , 
. Hence, we have proved our claim by induction. Now, raising both sides to the 
th power,
So,
and
Are these identities useful? And can you use them to compute the sum?
is on the curve 
. There is a parabola 
with focus . Two tangent lines to
are drawn through a point 
on 
, and the tangent points are 
and 
respectively. The line 
parallel to the line is tangent to at point 
. Question: When the line and 
have two intersection points, find the range of 
.\ as 
.\
, then 
, so 
,and 
.\
.\ with the curve to get an equation and then uses Vieta's theorem.
The answer is that according to the question, the equation has two negative roots. I can't understand this, and this is exactly where my problem lies.\
.
.
be a circle inscribed inside a rhombus 
. Let and be variable points on and respectively, such as 
is always the tangent line to . and the area of triangle 
is the same.
Prove that
and 
.
Sorry, This is the first time I post so I don't know how to text as mathtype.
which is not so hard.