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.
is divisible by 
for all positive integers 
.
and satisfying 
.
be a polynomial in one variable with integer coefficients. Prove that the number of pairs 
of positive integers such that 
, is finite.
be a cyclic quadrilateral. Assume that the points 
are collinear in this order, in such a way that the line 
is tangent to the circle 
, and the line 
is tangent to the circle 
. Let 
and 
be the midpoints of segments 
and 
, respectively. Prove that the following three lines are concurrent: line 
, the tangent of circle 
at point 
, and the tangent to circle 
at point 
.
+1 w
be nonnegative real numbers such that 
. Let 
. Prove that 
and determine when the equality holds.
from the nonnegative integers to the integers satisfying 
and ![\[ f(mn) = f(m) + f(n) + f(m)f(n) \]](http://latex.artofproblemsolving.com/e/f/f/eff10a32cffaec742403ebfc46ff09bc8ef7b7c5.png)
for all nonnegative integers and .
, 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 point in the plane of triangle 
such that the segments 
, 
, and 
are the sides of an obtuse triangle. Assume that in this triangle the obtuse angle opposes the side congruent to . Prove that 
is acute.
and be two distinct rays not lying on the same line, and let 
be a circle with center 
that is tangent to ray at 
and ray at 
. Let 
be a point on segment 
. The line through parallel to intersects line at . Let be the intersection of lines 
and , and let be the intersection of line and the line through parallel to . Prove that line 
is tangent to . be an acute triangle with point 
lie on the plane such that 
is a parallelogram. 
is the orthocenter of 
intersects at 
and 
intersects at 
The circle with diameter 
intersects the circumcircle of 
again at 
Prove that: The circumcircle of 
passes through the reflection point of over 
with 
, let 
denote the incenter and the midpoint of side . The line through perpendicular to 
intersects the tangent from to the incircle (different from line ) at a point > Show that is tangent to the circumcircle of triangle 
.
and 
. At each step, it is allowed to add to one of the polynomials and to multiply another one by the polynomial 
. Can the polynomials become equal at some point?
such that for any positive integer number 
and any arrangement of the numbers from 1 to 
on a circle, there exists a pair of adjacent numbers whose ratio (when dividing the larger number by the smaller one) is less than .
make the equation ![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
and we want to show it for 
. Take two groups, and , s.t that subset is in group iff it has in it. We can bound 
with inductive hypothesis, and |B| by using sperner. This way we are done.
such that no one is contained in another and no two cover whole 
?
,
satisfy 
.
.
with more than 
elements, rip me 
maybe a construction from here, but I couldn't get it.
,
contains an 
-
,
(indeed, each of them has 
-
times.) Replace our 
,
-
of them, and 
-
of them. Assume that 
.
such 
such 
,
,
.
we get an absurd 
.
since 
,
.
,
.