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 pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
such that 
is a perfect square.
and 
touches each other externally at a point 
, quadrilateral 
is inscribed in , and the lines 
, 
are tangent to at points 
and 
respectively. Line 
bisects 
and meets segment 
at 
. Line 
meets the arc 
(not passing through the point 
) at another point 
different from 
. Prove that is the circumcenter of 
.
points 
on the 
-plane.
. Denote by 
the distance between 
and the line 
. Let 
.
in terms of 
.
moves in the range of 
, express the maximum and minimum value of in terms of 
.
, define a function 
such that the inequality :![\[f(x) \leq f(x)^3-2f(x)^2+2\]](http://latex.artofproblemsolving.com/1/2/c/12c4c4c850c1c2886f002dfcb0bead075e2ab679.png)
.
of a sequence 
of positive integers satisfying both the following properties:
, and
(where 
) with a choice of signs 
for which ![\[s_ia_i+s_{i+1}a_{i+1}+\dots+s_ja_j=0.\]](http://latex.artofproblemsolving.com/4/b/6/4b66fe235e9a56cc646dbd38aae72f8ddfeb77de.png)
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such that, for every positive integer 
the integer
is a multiple of 
(Note that 
denotes the greatest integer less than or equal to 
For example, 
and 
)
be an acute triangle with circumcenter 
and orthocenter 
, satisfying 
. The tangent line at to the circumcicle of intersects 
in . Let 
be the midpoint of 
. Prove that 
.
and 
Prove that
Solution:
and 
then 
and be positive integers greater than 
. In each unit square of an 
grid lies a coin with its tail side up. A move consists of the following steps.
square in the grid;
for which it is possible that every coin shows head-side up after a finite number of moves.
be a positive integer. Prove that there exists a positive integer 
such that exactly 
of the integers 
are squarefree.
be the number of squarefrees among 
. Notice that 
. Pick sufficiently large primes 
. By Chinese Remainder Theorem for 
for 
, we see that 
. If there exists 
then there is some 
. Suppose that 
for each positive integer 
. Since density of squarefree numbers is 
we get a contradiction.
, everything else is as bin_sherlo. to divisible by 
for some prime 
is at most 
. Hence the amount of non-square free does not exceed 
by telescoping 
, so indeed the amount of squarefree is greater than 
.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
