AoPS Academy
be a triangle with orthocentre 
. Let 
be the foot of altitudes of 
onto the opposite sides, respectively. Consider 
, the reflection of 
about line 
. Let line 
cut at distinct points 
, and let 
be the orthocenter of 
. Prove that points 
are concyclic.
be a continuous function such that 
show that 
is even.
grid, the number 1 is written. A move consists of choosing a positive integer 
and a cell, and then multiplying the numbers in that cell and its neighbors by . Is it possible, after a finite number of moves, for every cell on the grid to contain the number 
?
+2 w
is a minuan pair if it satisfies the following two conditions:
is even.
are all the positive divisors of , ordered such that 
, then the set of all positive divisors of 
is precisely
.
, with two of the points, say 
in its interior.
meet two side of the triangle (recall that no three points are collinear), say 
and 
. Thus 
are in some order the vertices of a convex quadrilateral, and we are done.
points be 
, 
, 
, 
, and 
. We first consider , , , and . If each of these is in the exterior of the triangle determined by the other 
points, we are done by 
. If one of these is in the interior, however, WLOG let it be . We now proceed with casework based on the position of . is in the exterior of triangle 
, the lines , , and 
split the plane into regions. If is in the region determined by and , then 
is convex, and analogous reasoning applies to the other 
regions. is in the interior of triangle , extend 
bast to intersect 
at , and define 
and 
similarly. Then, if is in the interior of triangle 
or 
, 
or 
is convex respectively. Analogous reasoning applies to the other pairs of triangles. 
points forming a convex quadrilateral.
