[/quote]
,
such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight?
be nonzero real numbers such that 
.
Show that for any odd integer 
,
are written on the board.In each step,two random numbers 
and 
are chosen and deleted.Then,the number 
is written instead.What will be the number remained on the board after the last step.
,![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
is satisfied. Point 
is taken on the circumcircle of 
such that 
.
which passes from 
and 
respectively at 
and 
.
passes from circumcenter of 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
.
,
point on AC, AB respectively such that 
,
.
,
lines intersect at point 
.
,
be the unit circle in the complex plane. Let 
be the map given by 
.
and 
for 
.
such that 
is called the period of 
.
of period 
.
)
and let 
be positive integers such that 
.
and determine when equality holds.
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
prove that the triangle must have a right angle.
be six positive real numbers. Prove that
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
Y by
A line through the vertex 
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
of the triangle 
which doesn't coincide with 
or 
intersectes the altitudes from 
and 
at 
and 
respectively. Let 
be the reflection of in and 
be the reflection of in 
Prove that the circles 
and 
are tangent.
be intersection of 
and 
.
.
is cyclic. We can also see that 
,
,
have common tangent.
be foot of perpendicular line of C. Then 
done.
