[/quote]
,
I
be positive real numbers, show that:
such that 
and 
are also primes.
be pairwise distinct integers. Show that the equation 
has at most one integral solution.
be a positive integer and let 
be odd primes. Prove that the positive integers 
can be colored in 
colors, such that for any 
of the same color, 
is not divisible by 
.
be its circumcenter and 
its incenter.
be the radical center of its three mixtilinears and let 
be the isogonal conjugate of 
be the Gergonne point of the triangle 
is parallel with line 
.
be a polynomial with real coefficients such that 
for all 
.
for some real constant 
and 
.
= 2∠AMP.
.
where 
are integers in 
are each labeled with a positive integer less than or equal to 
such that every label is used exactly 
and 
such that the following property holds: For any two lattice points 
and 
that are both labeled 
,
such that 
and 
are both divisible by 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
or 
.
and 
,
and 
operated by two different companies, then there must exist flight route 
operated by the third company different from 
,
and 
denote the number of flight routes operated by companies 
.
such that![\[f(x^{n+1}+y^{n+1})=x^nf(x)+y^nf(y),\forall x,y>1.\]](http://latex.artofproblemsolving.com/a/4/7/a47d53806f2148cc3339335fb990b33641085896.png)
be a fixed integer. Find the least constant ![\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]](http://latex.artofproblemsolving.com/d/d/e/dde2d1dd0c412d8ba640f1d4c044e2fa2dcccc82.png)
(the sum on the left consists of 
summands). For this constant 
,
be the least prime that does not divide 
,
to be the product of all primes less than 
For 
,
.
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
.
.
Y by Adventure10, Mango247
Let 
be the midpoint of side 
of triangle 
(
), and let 
be the bisector of the angle 
. The line passing through perpendicular to intersects the side 
at the point 
. Prove that 
is equal to half the perimeter of triangle .
be the midpoint of side 
of triangle 
(
), and let 
be the bisector of the angle 
. The line passing through perpendicular to intersects the side 
at the point 
. Prove that 
is equal to half the perimeter of triangle .
so we need to prove that 
.
as the point on side 
(and hence 
)
i.e. that 
.
.
as the point such that 
.
.
so 
hence 
is isosceles.
as required.
cut resp. 
at 
it s clear that 
are isoceles and 
where 
.Since 
pass through the midpoint of 
are the midpoints of 
hence 
Then, by LoS and 
isosceles, 
Hence, 
