[/quote]
,
be a triangle with incenter 
,
be a point on side 
and 
are chosen on lines 
and 
respectively such that 
is a parallelogram. Points 
and 
are chosen on side 
and 
are the angle bisectors of angles 
and 
respectively. Let 
be the circle tangent to segment 
,
past 
past 
be an even prime number and a sequence 
satisfy 
and 
Prove that always exists positive integer 
satisfying for all positive integers 
,
.
is "not congruence"
![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
be pairwise coprime positive integers with 
being prime and 
.![$I = [0, a_1 a_2 \cdots a_n ]$](http://latex.artofproblemsolving.com/a/2/b/a2bce96b048c9b8fc94926db80aba37fd5037b4a.png)
of the real line, mark all integers that are divisible by at least one of the numbers 
. These points split 
units horizontally (left or right); or
and 
,
to 
using at most 
moves.
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
.
such that:
such that:
is given, 
are graphs on the same set of 
connected components respectively. Prove that 
has at least 
connected components.
and 
intersect at points 
and 
intersects 
,
intersects 
.
is the circumcenter of 
,
.
and a subset of the numbers written on the board and adds 
be real numbers . Prove that : There exist positive integer 
such that 
Where 
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
and 
be positive integers.
and 
.
.
Y by Adventure10, Mango247
A friction-less board has the shape of an equilateral triangle of side length 
meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex 
towards the side 
. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form 
, where 
is an integer. What is the value of ?
meter with bouncing walls along the sides. A tiny super bouncy ball is fired from vertex 
towards the side 
. The ball bounces off the walls of the board nine times before it hits a vertex for the first time. The bounces are such that the angle of incidence equals the angle of reflection. The distance travelled by the ball in meters is of the form 
, where 
is an integer. What is the value of ?
i think so answer is 31
