[/quote]
,
be the center of the square inscribed in acute triangle 
with two vertices of the square on side 
.
and the other is on 
.
are defined in a similar way for inscribed squares with two vertices on sides 
are concurrent.
and 
be circles with centres 
and 
,
and 
.
intersects 
and 
,
lie on 
be the circumcentre of triangle 
.
meets 
and meets 
.
be the orthocentre of triangle 
.
.
if it is not parallel to any of the 
–
–
.
be a given integer. Determine all nonnegative integers 
such that there exist 
distinct lines in the plane satisfying both of the following:
and 
with 
,
lies on at least one of the lines; and
denote the set of positive integers. A function 
is said to be bonza if![\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]](http://latex.artofproblemsolving.com/7/b/b/7bb6b2b4fcde900ee6f7ef90b906525d61312796.png)
for all positive integers 
such that 
for all bonza functions 
and all positive integers 
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set 
.
be the circumcenter and 
,
and 
is equal to the sum of the areas of the other two.
.
and 
,
be the point on line 
be the point on line 
intersect the circumcircle of triangle 
.
and 
be the midpoints of 
.
be a quadrilateral with 
,
,
,
,
the second intersection of 
and 
,
and 
;
.
,
the radius of the incircle of triangle 
and 
,
is tangent to the circumcircle of triangle 
the circle through 
,
the circle through 
.
,
,
,
,
.
meets the incircle at two distinct points 
and 
,
meets the incircle at two distinct points 
and 
.
and 
passes through the midpoint 
.
,
,
and 
and 
is equal.
and 
lie on 
and 
.
meet 
meet 
and 
have an intersection lie on line 
.
equal sectors. On each circle 
sectors are painted white and the other 
Y by Rajukian
Define a "hook" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
Determine all mxn rectangles that can be covered without gaps and without overlaps with hooks such that
- the rectangle is covered without gaps and without overlaps
- no part of a hook covers area outside the rectangle.
Determine all mxn rectangles that can be covered without gaps and without overlaps with hooks such that
- the rectangle is covered without gaps and without overlaps
- no part of a hook covers area outside the rectangle.
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