[/quote]
,
the feet of the altitudes lying on 
respectively. One of the intersection points of the line 
The lines 
and 
meet at point 
Prove that 
be positive integers such that ![$\sqrt[n]{a}+\sqrt[n]{b}$](http://latex.artofproblemsolving.com/d/a/a/daaf95cd3381060559529aa4a318e97b727eac31.png)
is an integer. Prove that 
are both the 
t
positive integers.
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.
such that 
is a prime number and 
.
is irreducible over ![$\mathbb{Z}[x]$](http://latex.artofproblemsolving.com/4/b/7/4b7d768fffbb7a1a6163af392ed348f2d49d8783.png)
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
satisfies the conditions![\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]](http://latex.artofproblemsolving.com/5/5/6/556f545540d4941d058a1c4099b4b86eb35caf0d.png)
for all real 
.
such that
for all 
.
,
of rational numbers, where 
are relatively prime positive integers for each 
such that the positive integers 
are all distinct.
![\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]](http://latex.artofproblemsolving.com/3/d/f/3dfb1d52835627a2977f0074f8df7e7ed677f2a5.png)
can be written as the product of one or more elements of 
of integers with 
,
,![\[f(xy)<f(x)+f(y).\]](http://latex.artofproblemsolving.com/c/f/f/cff5554b8e72d99f8989448c2621794242d784d8.png)
(Pairs 
and 
are different if 
or 
)
denote the set of natural numbers, and let 
,
,
.
whose product is a perfect cube.
prove that the triangle must have a right angle.
,
and 
intersect at 
.
and 
intersect again at 
. If 
, show that 
.
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is 
Y by Akacool, IMUKAT, ohhh
Let 
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, 
, and 
-median of 
are concurrent.
,
,
of 
belongs to 
(which I don't even use in the solution, but it's what gives us the initial path to simplify the problem).
,
and 
,
coincide, but the centers of the first two don't seem too bad to work with,
and the center of 
.
!
is perpendicular to the radical axis 
of 
unpretentiously, we see that if 
!
.
and the projections that I take in lemma 
.
on the line 
on the line 
and 
on the line 
)
with the circle 
,
of 
is in 
.
and 
almost disjoint lemmas, that is, solve 
Lemma 
