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.
with 
. Points 
and 
are on 
and 
respectively such that 
. Lines 
and 
meet at 
. 
is the incenter of 
and 
is the orthocenter of 
. 
and 
are circles with diameter 
and 
, respectively, intersecting each other at points 
and 
. Prove that and lie on 
.
and 
are externally tangent at a point 
. Let 
be a line tangent to at 
and at 
. Let 
and 
be diameters in and respectively. Suppose points 
and 
lies on such that 
and 
are tangent to , and points 
and 
lies on such that 
and 
are tangent to ., , , lie on a circle 
., , , determine a quadrilateral with an incircle 
, and its radius is 
times the radius of .
and respectively be the incentre, circumcentre and circumcircle of triangle . Points 
are chosen on , such that 
, and point 
is the foot of the altitude from to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
and 
for which:
, 
be positive integers such that 
. Suppose 
is a sequence of integers satisfying 
, 
, and for all indices 
, 
Prove that there exists 
such that 
and be positive integers. Find all pairs of non-negative integers 
and 
that always satisfy the following condition: white dots and black dots on a circle, there always exist a line cutting the circle into two arcs, one of which consists of exactly white dots and black dots.
such that 
, find the maximum possible number of integers 
such that 
with degree 
, such that for all positive reals 
with product 
, then; 
Suppose there exist positive reals 
with product , such that; 
Is it possible that the values are all distinct?
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
are positive real numbers such that 
then 
.
![\[
\textbf{Problem:}
\]](http://latex.artofproblemsolving.com/5/c/3/5c311b6a956843df04ab5bdf4117e1089aecb9b9.png)
be the roots of the polynomial![\[
P(x) = x^{20} - 7x^3 + 1.
\]](http://latex.artofproblemsolving.com/1/8/c/18c475088b173b3cb3bf72d1d83ef8f2d40b8e36.png)
![\[
S = \frac{1}{r_1^2 + 1} + \frac{1}{r_2^2 + 1} + \dots + \frac{1}{r_{20}^2 + 1}
\]](http://latex.artofproblemsolving.com/b/d/6/bd63906510edd807858482b9c7ca3ca202da52d6.png)
, where 
and 
are coprime positive integers. Determine the value of 
.
.
satisfying:
and let 
and 
,
and 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
![\[
19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100
\]](http://latex.artofproblemsolving.com/d/3/d/d3d8657e2ab69cdc0886e2498ab03f7fd959056f.png)
times the number of its sides. How many sides does it have?
diagonals. From the given, we have 
,
gives 
,
,
.
.
,
.