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.
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
be a function such that
for all 
. Determine all such functions 
.
such that the equality 
holds for all pairs of real numbers 
.
with integer coefficients such that 
and 
is a square of an integer for all nonnegative integers 
.
and an integer 
, 
is defined as follows: 
if 
and 
if 
.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
and 
be points on a plane such that 
, where is a positive integer. Let 
be the set of all points such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that for all , is always a rational number.
be a positive integer. A positive integer is called a benefactor of if the positive divisors of can be partitioned into two sets and such that is equal to the sum of elements in minus the sum of the elements in . Note that or could be empty, and that the sum of the elements of the empty set is 
.
is a benefactor of 
because 
. has at least 
benefactors.
denotes the least natural such that
Find all naturals 
such that 
.
square is divided into unit squares. Is it possible to fill each unit square with a number 
in such a way that, whenever one places the tile so that it fully covers nine unit squares, the tile will cover nine different numbers?
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be a natural number. The numbers 
are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the 
GCDs obtained?
be reals such that 
. Prove that
Let be reals such that 
. Prove that
and 
. Then we should find this ratio: 
.![\[S_{EGC}=(a+x)^2+\frac{x^2}{2}-\frac{(a+x)(a+2x)}{2}-\frac{a(a+x)}{2}=\frac{x^2}{2}.\]](http://latex.artofproblemsolving.com/2/a/7/2a7271b3023b31cbdadac9161187b92f1a292986.png)
![\[S_{DEFBC}=(a+x)^2+x^2+\frac{ax}{2}.\]](http://latex.artofproblemsolving.com/c/f/0/cf022939edf4bba3c1ce126b096a1e193665deee.png)
![\[S_{DEFBC}=16S_{EGC}\Rightarrow a^2+2.5ax-6x^2=0\]](http://latex.artofproblemsolving.com/e/a/6/ea6e9085f8aac29b7c60539cc726c5be3c33a48e.png)
![\[\left(\frac{a}{x}\right)^2+2.5\frac{a}{x}-6=0,\quad \frac{a}{x}>0\Rightarrow\frac{a}{x}=\frac{3}{2}.\]](http://latex.artofproblemsolving.com/f/3/0/f3047f7e354ad1e790923424d1205422532724df.png)
Therefore,![\[\left(\dfrac{x}{a+x}\right)^2=\left(\dfrac{1}{\frac{a}{x}+1}\right)^2=\frac{4}{25}.\]](http://latex.artofproblemsolving.com/9/3/8/9388ead9972342deafa8461cdcb8961e91254ed0.png)
Hence answer is: 
.
is an obvious solution. Now, let 
.
or 
.
.
lies on side 
of square 
and square 
is drawn outwards 
is 
of the area of pentagon 
.
