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.
where 
. Point 
is the circumcenter of triangle , and 
is the projection of point 
onto line 
. The midpoints of , 
, and 
are 
, 
, and 
, respectively. The line 
intersects 
and 
at points 
and 
, respectively. Prove that is the incenter of triangle 
.
) with the school (point 
). In the square there are 
pupils starting to go to the school. They have the ability to move only to the right and up. If the pupils are free to choose any allowed path (in order to get to school), determine the minimum value of so that in any case at least two pupils follow the same path.
be real numbers satisfying![\[x^2 +y^2+z^2 = 1 \]](http://latex.artofproblemsolving.com/d/1/8/d18e31dce3f312d786cca935d497743ba76091ef.png)
Prove that![\[ Q = xy + yz +2xz \le \dfrac{1+\sqrt{3}}{2}\]](http://latex.artofproblemsolving.com/e/5/6/e563c1223d410c64506a55e1ced878df96b7971c.png)
, such that 
for any positive reals 
.
, a set is said to be -good if the set contains exactly three elements 
and 
. such that 
can be partitioned into several -good sets.
be an integer. Consider 
squares with side lengths 
, respectively. The squares are arranged in the plane with their sides parallel to the 
and 
axes. Suppose that no two squares touch, except possibly at their vertices. Show that it is possible to arrange these squares in a way such that every square touches exactly two other squares.
without using the following method. I know that you can solve it using 
but I'm searching for an algebraic proof.
is integer, find the possible values of 
.
is a triangle, 
,
are the midpoints of 
, 
. 
be the orthocenter of 
. 
and 
. Prove that 
and 
are parallelograms.
be the incenter, centroid and the circumcenter of triangle , respectively. Let 
be on the rays 
respectively so that 
. Let 
be the centroid of 
.
is perpendicular to 
.
different positive integers , less than 
, there exist some two which are NOT relative prime. such that 
is a power of 
.
diagonals intersect at 
and 
Let be the foot of the perpendicular from 
to 
and be the second intersection point of the circumcircle of triangle 
and line segment 
. Find 
.
be the foot of perpendicular from 
onto ; from the given segments relation, 
, since 
is right-angled and isosceles. Hence 
.
are altitudes, so 
, while p.o.p. 
w.r.t. 
gives 
, consequently 
is cyclic, and 
.
be the foot of perpendicular from onto ; from the given segments relation, , since is right-angled and isosceles. Hence . are altitudes, so , while p.o.p. w.r.t. gives , consequently is cyclic, and .
. So 
??
divided by 
, such that 
Prove that :
