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.
such that the largest prime divisor of 
is equal to the largest prime divisor of 
.
and 
Show that
Where 
and 
Show that
Where
: 
such :
. She then multiplies this number by and adds 
to the result. Pat will add 
to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by and add to the result when it is her turn. Pat will always add to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by 
. Mary wants to prevent Pat from winning the game.
with at most four natural divisors, which have the property that for any two distinct proper divisors 
and 
of , the positive integer 
divides .
is a positive integer. Call all positive divisors of which are different from 
and beautiful divisors.We call a special number when it has at least beautiful divisors and difference of any beautiful divisors divides as well. Find all special numbers.
, where 
and 
are (not necessarily distinct) positive integers. The integer 
is written as the sum of 
Tiranian integers. What is the smallest possible value of ?
be non-nagative real numbers such that 
.
.
and 
Prove that 
Let 
and Prove that 
Let and 
Prove that 
and its internal angle bisector 
. The line intersects the circumcircle 
of triangle at 
and 
. Circle 
with diameter 
cuts again at 
. Prove that 
is the symmedian line of triangle .
be a triangle with circumcircle 
. Arbitrary points 
on 
respectively. Circumcircle 
of triangle 
intersects at 
. 
intersects 
at 
. 
cuts and at 
respectively. Construct parallelogram 
. Prove that 
are collinear.
be a triangle, and let 
be a point inside it such that 
. The perpendiculars from to 
and 
meet these lines at 
and 
, respectively, and 
is the midpoint of 
. Prove that 
and use cosine rule in 
to find out 
.Now its easy to verify that its equal to 
.
.
and 
, 
and 
respectively. obviously 
and 
.
so: 
and so 
.
respectively,then prove that 
=