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.
be the circumcircle of triangle 
, 
is the midpoint of 
and 
be the second intersection of 
and . Tangent line of at intersects at 
, let 
be a transversal of and 
be intersections of 
with . Let 
be a tangent line of . Prove that 
is cyclic
be positive integers that satisfy the inequality![\[
\frac{a}{b} < \frac{x}{y} < \frac{c}{d}
\]](http://latex.artofproblemsolving.com/8/d/9/8d982d52e655546ac29039fa6e440e68821bfe7f.png)
with 
. If 
, determine the minimum value of 
.
. Let 
and be the feet of the altitudes from 
to and from 
to 
respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
be a triangle whose incircle is tangent to 
, 
, 
at 
, respectively. Let and 
be the midpoints of 
and 
, respectively. Suppose that points 
and 
lie on 
so that 
and 
. Prove that lines 
and 
intersect on the circumcircle of 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points and lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
be a triangle, and let be on so that 
. Let 
intersect the circumcircle of again at . Let 
be the midpoint of 
and let the perpendicular bisector of 
intersect 
at 
. If is the circumcenter of 
, prove that 
.
be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form 
, where 
and 
are not necessarily distinct. Determine 
.
be a cyclic quadrilateral with 
. Let be the midpoint of the arc 
not containing . Suppose there is a point inside such that 
and 
.
, and are concurrent.
. By one question, we may take three different indices 
and find out the set of numbers 
(unordered, of course). Find the minimal number of questions, which are necessary to find out all numbers 
.
be an equilateral triangle with side length 
and be a point inside the triangle such that 
. What is the area of a triangle with side lengths 
?
be the number of positive integers that divide the integer 
. For all positive integral divisors of 
, what is the sum of numbers 
?
such that the value of the
is a square of an integer.
be real numbers such that 
for 
.
b) Does there exist a real number such that 
for all real numbers 
such that for .
, thus 
. So the minimal value is 
which is achieved when 
and 
for all 
. Then the value is 
which can get arbitrarily small, thus such does not exist.
, then 
.
:![\[ -2a_ia_{i+1} \leq a_i^2 + a_{i+1}^2 \leq 1+2a_ia_{i+1} \implies a_ia_{i+1} \geq -\frac{1}{4} \]](http://latex.artofproblemsolving.com/8/8/a/88a063e1950152f50449fe297728b9e6c9ffdc2e.png)
. Equality is reached when 
and 
.
for 
and 
. The expression becomes 
. This becomes arbitrarily negative, meaning such a can't exist. Q.E.D.
is 
when 
or the other way round. This makes the minimum value of the sum 
for 
and 
only appear once. Thus, we let 
where the equation will give us 
as the result. Since the result of the equation depends on