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.
are positive real numbers satisfied 
. Prove that:
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be a square with sides length 
. Let 
be a path within which does not meet itself and which is composed of line segments 
with 
. Suppose that for every point 
on the boundary of there is a point of at a distance from no greater than 
. Prove that there are two points 
and 
of such that the distance between and is not greater than 
and the length of the part of which lies between and is not smaller than 
.
and the sets 
If 
has exactly three elements, prove that 
cannot be an integer.
be a positive integer not divisible by 3. Prove that there are infinitely many positive integers such that 
and 
are divisible by , where 
is the sum of digits of 
.
books are given as prizes to students (each students gets one book). In how many ways can the organisers give these prizes if they have copies of one book and 
other books each in one copy?
. Show that there exists 
such that![\[a_1x_1^2+a_2x_2^2+\ldots +a_nx_n^2\geqslant (a_1x_1+a_2x_2+\ldots +a_nx_n)^2.\]](http://latex.artofproblemsolving.com/4/d/2/4d269e5b682be6778e89237aa91da5e0399042a7.png)
so that the sum of its edges' lengths is minumum.
find, in terms of , the minimum value of 
for which there is a set of 
distinct positive integers that has sum greater than but every subset of size has sum at most 
with 
equilateral. Let be the area of the triangle of sides 
, 
and 
. Show that 
where 
is the circumradius and 
is the volume of the tetrahedron. be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at . The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
where 
. such that for every 
we have 
. is irrational then for every 
there exists 
such that![\[u<P(n)-\lfloor P(n)\rfloor <v.\]](http://latex.artofproblemsolving.com/9/4/a/94a8cb1f064e1dea37e8894fb4fd4e34c4c6c845.png)
Generalize the problem!
we consider a variable point 
which moves on line 
towards 
. Halfway there, it stops and starts moving in a straight line line towards 
. Halfway there, it stops and starts moving in a straight line towards 
, and halfway there it stops and starts moving in a straight line towards , and so on. Show that will get as close as we want to the vertices of a fixed triangle with area 
.
denote the sum of the digits of a positive integer 
,