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.
. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if is odd; otherwise Adamu plays first. Prove that: is odd. is even.
that satisfies![\[ f(x+yf(x))f(y)=f(x)(1+yf(y+1)) \]](http://latex.artofproblemsolving.com/f/7/a/f7a138fd2dc76f034525491a10f62f1e83de294d.png)
for all 
.
, 
be positive integers such that 
is a multiple of 
for all positive integers . Prove that 
.
with 
is called historic if 
. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
satisfying 
for all pairs of positive reals 
and 
. Here, 
denotes the set of all positive reals.
be positive real numbers such that : 
. Prove that : 
Let 
be positive reals numbers such that 
.Prove that 
satisfying: 
, 
and the function 
defined by![\[f(n)=\left\lfloor \frac{an+b}{cn+d}\right\rfloor\text{ for all } n\in\mathbb{Z}_{\ge 0}.\]](http://latex.artofproblemsolving.com/8/f/b/8fb372df4b44c3d7c0690b310b12c684134fee74.png)
Prove that the following are equivalent:
is surjective;
, 
and 
.
and 
be on sides 
and 
respectively in 
such that 
. Let 
be the circumcenter of 
and let 
. Prove that 
+1 w
, where 
is the set of real numbers, such that
be the n-th prime number in increasing order for 
. Prove that there exists a sequence of distinct prime numbers 
satisfying 
for all 
is an interior point of a triangle 
such that the ratios ![\[ \frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)} , \frac{d(C,AB)}{d(P,AB)} \]](http://latex.artofproblemsolving.com/1/c/1/1c19c15ec63b0a94206e8829c07f93d6aa4546bc.png)
are all equal. Find the common value of these ratios. 
represents the perpendicular distance fro 
to the line 
.
![\[ \frac{(1/2) a d(A,BC)}{(1/2) a d(P,BC)} = \frac{(1/2) bd(B,CA)}{(1/2) bd(P,CA)} = \frac{(1/2) cd(C,AB)}{(1/2) cd(P,AB)}= \frac{3\Delta}{\Delta}=3, \]](http://latex.artofproblemsolving.com/6/6/f/66f995a47c8cbdb111dc53ceee8f4c3988db90af.png)
where the last equality follows from summing the numerators and denominators.