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 fixed points on this order on a line. Let 
be a variable circle through 
and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
. Let 
and 
be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
of positive integers such that 
is multiple of 
and 
is multiple of 
.
be an acute triangle and let 
be an interior point for it such that 
. Prove that
When does equality hold?
be non-negative number. Prove that 
+1 w let 
be the 
-excenter. Let be an arbitrary circle that passes through 
and intersects the extensions of sides 
(extended from 
) at 
respectively. Let 
be points on segments 
respectively such that 
and 
.Lines 
intersect at 
. Lines 
intersect at .
are collinear.
be a convex quadrilateral with 
. Let 
be the point of intersection of 
with 
and let 
be the midpoint of 
. On the extension of , beyond the point , we pick a point such that 
. Let 
and 
be the projections of and respectively on 
. The circumcircle of the triangle 
meets again at the point 
. If 
is the point of intersection of with , prove that the lines 
and 
intersect on the line 
.
. Let 
be the circumcenter of this triangle . Let 
, , be the feet of the altitudes of triangle from the vertices , 
, , respectively. Denote by 
, 
, 
the midpoints of these altitudes 
, 
, 
, respectively. The incircle of triangle has center and touches the sides , 
, 
at the points , , 
, respectively. Prove that the four lines 
, 
, 
and 
are concurrent. (When the point concides with , we consider the line as an arbitrary line passing through .)
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities and we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values . Consider a 
grid, a set 
of squares is called connected if for any points 
in , there exists an integer 
and 
squares 
in such that and shares a common side (
). satisfying that however the squares are colored black or white, there always exists a connected set for which the absolute value of the difference between the number of black and white squares is at least .
be an acute triangle and 
its circumcircle. Let be the line tangent to at . Let and be the intersections of the circumference with center and radius with lines and 
, respectively. Prove the orthocenter of lies on line 
.
be a cyclic quadrilateral an let be a point on the side 
The diagonals meets the segments 
at 
The line through parallel to mmets the extension of the side 
beyond at 
The line through 
parallel to 
meets the extension of the side beyond at 
Prove that the circumcircles of the triangles 
and 
are tangent .
be an integer greater than 
. For a positive integer 
, let 
. Suppose that there exists a 
-element set such that is an -element subset of 
; shares at most one common element; is contained in exactly two elements of . in terms of .
is ![$\left[\sqrt{\alpha n} + \frac{1}{\beta}\right]$](http://latex.artofproblemsolving.com/1/2/b/12b45d9d0f5268002da2adae8dac73390a106d96.png)
(Here ![$\left[.\right]$](http://latex.artofproblemsolving.com/3/c/a/3ca27308567a179890adebe5721a274c19ace40f.png)
denotes GIF) then
Let 
and 
then the number of numbers out of the first 
natural numbers which are divisible by 
or 
is
None of these
Let 
and 
. The number of divisors of the number 
which are of the form 
is equal to
None of these
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
squares of a board be colored red. This colouring will be called a k - saturation if and only if coloring any one of the remaining squares red will result in a block of 
squares at least 
of which are red. Find the least such that a k - saturation exists.
red in some 
board . Suppose otherwise. Note that we cannot have more than red squares in every 
adjacent columns. Since we have 
columns, we can have at most 
red squares. Contradiction.
, we cannot have 
or more red squares unless it is of the following configuration:
or more red squares. In a sub-board with rows and columns, suppose we can find two adjacent red squares in a row, then consider these two columns with either the immediate right column or left column. We can have at most red squares in these columns. Hence we have at least 
left for 
columns. Grouping them into blocks of , we have a contradiction. Hence we cannot find two adjacent red squares in a row, and hence it must be
red, we have 
left for 
rows. Contradiction. If not, remove the top rows. We will be removing at most red. Repeat this process to get a contradiction.
sub-rectangle of the 
square. It is made up of overlapping 
block. The maximum number of red squares in the 
rectangle is because if we choose any red squares, it is easy to show that of them are part of square. Hence, since there are 
non-overlapping rectangles of this configuration, with a total of 
red squares, it leaves row of uncolored squares at the bottom. All of these are not part of any considered 
square, and hence can be colored for a maximum of 
red squares.
red squares in 
is the best.