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.
sides are colored gold or silver. Prove that there are at least 
different isosceles triangles whose vertices have the same color.
such that 
. Prove that![\[\sqrt[4]{(1-a^4)(1-b^4)(1-c^4)(1-d^4)}\geq 255\cdot abcd.\]](http://latex.artofproblemsolving.com/d/e/6/de61611f628ea796c706d05941d7ee9b3fc22916.png)
, then either travels [list] units vertically (up or down) and 
units horizontally (left or right); or units horizontally (left or right) and units vertically (up or down)., the ant can choose to go to one of eight possible points.
and 
, the ant can travel from 
to 
using at most 
moves.
be a set of 
elements and 
are subsets of (
), such that every one of them has at least 
elements.
and 
, with 
, such that the number of common elements of 
and 
is greater or equal to: 
, the sides 
and 
are parallel and the angles 
and 
are acute. Show that it is possible to divide the triangle 
into 4 disjoint triangle 
and the triangle 
into 4 disjoint triangles 
such that the triangles 
and 
are congruent for all .
has side lengths , , and 
. Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
be non-negative real numbers such that 
Prove that
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of are either all odd or all even. is interesting if 2018 divides 
(the number of positive divisors of ). Determine all positive integers such that there exists an infinite arithmetic progression with common difference whose terms are all interesting.
of a right triangle , with 
, cut the angles bisectors 
and 
at and 
, and let 
and be the midpoints of 
and 
. Prove that 
is parallel to the hypotenuse of 
),
and 
are isosceles, so 
and 
. Let 
be the incentre of (the intersection of and ), then 
is obviously cyclic and 
. Hence 
.
is cyclic, so 
. So 
and lies on the perpendicular bisector of . also lies on this perpendicular bisector, so 
*IS* in fact the perpendicular bisector of , and therefore it is parallel to .
.
such that for all 
such that
