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.
be a nonempty set of positive integers such that, for any (not necessarily distinct) integers 
and 
in , the number 
is also in . Show that the set of primes that do not divide any element of is finite.
be an acute triangle with orthocenter 
and circumcircle 
. A line through intersects segments 
and 
at 
and 
, respectively. Let 
be the circumcenter of 
, and suppose line 
intersects again at a point 
. Prove that line 
and the line through perpendicular to 
meet on .
less than 
yields the primes 
. After that, it is clear that 
has to be even to yield an odd number.
. All of these transformations don't seem to help. Is there any theorem I can use? Or is there something I'm missing?
be a point on a circle 
, and let 
be a point distinct from on the tangent at to . Let 
be a point not on such that the line segment meets at two distinct points. Let 
be the circle touching at and touching at a point on the opposite side of from . Prove that the circumcentre of triangle 
lies on the circumcircle of triangle .
has at least distinct prime divisors.
, an arbitrary point 
is chosen on the side 
. In triangles 
and 
, the angle bisectors 
and 
are drawn, respectively. The point 
is the circumcenter of 
. Prove that the second intersection point of the circumcircles of triangles 
and 
lies on the line 
.
be a sequence of positive real numbers that satisfies 
and 
for every natural number . Prove that 
for every natural number .
of real numbers such that![\[ a_{m^2 + m + n} = a_{m}^2 + a_m + a_n\]](http://latex.artofproblemsolving.com/b/b/9/bb97053e86e7733fe5c26f7859c87566eb444b52.png)
for all positive integers 
and .
be an acute triangle with 
, midpoint 
of side 
, altitude 
(
), and orthocenter 
. A circle passes through points 
and 
, is tangent to line 
, and intersects the circumcircle of triangle at a second point 
. The circumcircle of triangle 
intersects line 
at a second point 
. Prove that the lines 
and 
are perpendicular.
such that for all positive real numbers 
,![\[x^2[f(x)+f(y)]=(x+y)f(yf(x)).\]](http://latex.artofproblemsolving.com/2/7/c/27c95af9b083a2a2827d428da24a5152fd926473.png)
Determine all sets 
containing 
and having the property that 
is prime, for all distinct 
We thus need to prove
After squaring, the latter writes as
We square this too and get 
which is 
Equality at 
i.e. 
which makes the angle from right. I have the strong belief we are talking about Tereshin here.
We thus need to prove
If 
or if then we clearly have equality. Otherwise, since
The proof is complete. Equality if and only if 
or 
be a point such that 
, 
perpendicular to , and and on different sides with respect to .
, 
, 
...(1).
be the reflection of with respect to point . From this, we have that 
is parallelogram 
and 
., 
, 
. 
...(2).
and 
, we can easily find that 
and 
...(3).
and we have that 
...(4).
. and , so we try to take advantage of that.
, defined just like in post #7. Furthermore, let 
be the point where the A-Symmedian meets 
. It is well known that lies on the A-Apollonian circle, so 
. This, combined with the fact that 
, gives that 
.
give that 
.
finishes the problem since 
.
and 
Prove that
?
.
and points 
to get an even stronger result