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.
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 
. Assume 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.
positive real numbers satisfying 
,show that:![$\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$](http://latex.artofproblemsolving.com/c/b/3/cb39e7f838c99ed8a209c6844f3d2810838d1dfd.png)
.
+1 w
is satisfied. Point 
is taken on the circumcircle of 
such that 
.A line parallel to 
which passes from intersects 
and 
respectively at 
and 
.Show that circumcircle of 
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 
.
triangle with 
. We take 
, 
point on AC, AB respectively such that 
, 
. 
, 
lines intersect at point 
. If 
, find 
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer 
such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
satisfy the equality, 
prove that the triangle must have a right angle.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
meet at then from Seva teorem
and 
be the feet of altitudes from 
, respectively.Let 
be feet of perpendiculars from to the side and to the altitude 
and let 
and 
be the feet of perpendiculars from 
and 
to the side , and foot of altitude to 
, respectively.
, and 
with ratio 
.
and 
is Nagel point of the triangle 
and by Nagel line theorem 
are collinear and 
, on the other hand 
are collinear ( Euler line ) and 
,so the triangle 
and the triangle 
similar, so 
parallel to 
and 
.
be six positive real numbers. Prove that
,
,
,
.
are the incentre, circumcentre and orthocentre of 
respectively. Point 
,
and 
,
.
and 
be 
and 
.
,
,
.
is centroid of the trianle