[/quote]
,
be a circle inscribed inside a rhombus 
.
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to 
is the same.
,
such that any graph with 
vertices and at least 
passing through point 
is an oblique strophoid.
.
and 
externally on the sides 
and 
respectively.
meets 
at 
,
meets 
at 
and 
denotes circumcenter of 
and 
meets again on 
.
be a positive integer. Show that any polynomial ![$Q\in\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/e/8/2/e8286b6f2b9790f39c88849a1415d86f85c0b92c.png)
with all positive coefficients such that
is a perfect square for any 
must be a constant polynomial.
be an acute triangle and let 
be the foot of the 
be the midpoint of 
with diameter 
meets 
at 
,
with diameter 
at 
.
,
are concyclic.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
are taken such that the point 
.
and 
,
and 
.
.
Y by Adventure10
Each of 
identical jars contains some milk, up to percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other 
jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most moves.
(4 points)
identical jars contains some milk, up to percent of its capacity. At any time, we can tell the precise amount of milk in each jar. In a move, we may pour out an exact amount of milk from one jar into each of the other 
jars, the same amount in each case. Prove that we can have the same amount of milk in each jar after at most moves.(4 points)
pour from jar 
so after each move 
so we only need 9 move.
