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 orthocenter 
.Let 
and 
be the altitudes of triangle.Let 
and 
be reflections of points 
across the line 
, respectively.Let 
and 
be the midpoints of 
and 
, respectively.Let 
and 
.Prove that 
and 
are collinear.
be nonnegative real numbers such that 
. Prove that![\[
\frac{a}{4 - b} + \frac{b}{4 - c} + \frac{c}{4 - a} + \frac{1}{16}(1 - a)^2(1 - b)^2(1 - c)^2 \leq 1,
\]](http://latex.artofproblemsolving.com/5/b/d/5bd3349071e075519bd986c845c500125b7d46f8.png)
and determine all such triples 
where the equality holds.
let be the midpoint of 
. Let 
be a circle inside of and is tangent to 
at , respectively. The tangents from to meet at 
such that 
and 
lie on the same side of 
. Let 
and 
. If 
prove that 
is tangent to . and 
be the incircle and circumcircle of the acute triangle , respectively. Draw a square 
so that all of its sides are tangent to , and , are both on . Extend 
and 
, intersecting at and 
, respectively. Prove that 
and 
intersects on .
be real numbers such that 
Prove that if 
is a positive constant, then
is not always true
in which 
and 
. Let 
be a point on the side and 
a point on the line such that 
. Prove that 
.
, 
, 
and 
there exists infinitely many positive integers 
for which 
and 
are not relatively primes.
or 
tile.
be an integer. Dominoes are placed on an 
board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some 
such that each row and each column has a value of 
. Prove that a balanced configuration exists for every , and find the minimum number of dominoes needed in such a configuration.
be a cyclic quadrilateral inscribed in circle whose diagonals meet at . Lines 
and 
meet at . Segment 
intersects at . Lines 
and meet at , and lines 
and meet at . Prove that 
and concur with the tangent to at .
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
.
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
Prove that
Let 
Prove that
be positive integers. Let 
,
,
be three roots of the polynomial 
.
,
,
,