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 a positive integer. In an 
table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an table such that there exists only one possible upright path not containing any colored cells.
of positive integers satisfying the following two conditions:
, if 
is in 
, then so are 
and 
; and contains an integer 
such that 
is not divisible by 
.
such that 
and 
are all perfect square.
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.
such that 
and 
.
has perpendicular diagonals. The perpendicular bisectors of the sides 
and 
meet at a unique point inside . Prove that the quadrilateral is cyclic if and only if triangles 
and 
have equal areas.
such that
events 
that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:
consider 
the sum of 
over all 
such that 
and are not compatible.
If there are multiple such events, delete the event with the minimal weight.
are 
be a triangle with 
. Let 
be the reflection of 
across the line 
and be the reflection of across the line . Let 
and 
intersect at 
. Prove that the orthocenter of triangle coincides with the circumcenter of triangle 
.
be real numbers such that 
and 
. Find the largest possible value of .
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.
, 
is the incenter. Line 
intersects circumcircle of at 
, and it is given that 
. 
are points chosen on such that 
. Prove that 
and 
.
and 
. Thus, we have 
and 
.
and 
. Subtracting these two yields 
.
and 
such that
.
be the feet of altitudes through 
respectively. Prove that 
![\[AE = \frac{AB}{2} \text{ [Sine Law]}\]](http://latex.artofproblemsolving.com/1/f/4/1f4b08c6948b1805ac103ee4731568428c70b4d5.png)
![\[AF = \frac{AC}{2}\]](http://latex.artofproblemsolving.com/6/6/b/66b2d342b88d52058ef258453a69e2d25ca39f0c.png)
![\[AC - AB = (\frac{AB}{2} + CE) - (\frac{AC}{2} + BF)\]](http://latex.artofproblemsolving.com/d/c/d/dcd61946d20f84d2d42d70b983d5999c8e907093.png)
![\[CF - BF = AC - AB - \frac{1}{2}AB + \frac{1}{2}AC = \frac{2AC - 2AB - AB + AC }{2} = \frac{3}{2}(AC - AB)\]](http://latex.artofproblemsolving.com/6/f/b/6fbb4ad8ffc27beaeb8b0fc30c5824f134d997b2.png)
,
