[/quote]
,
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
.
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
its orthocenter, 
its circumcenter, and 
its circumradius. Let 
,
across the line 
,
across the line 
.
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
is a prime number
is inscribed in circle 
.
its 
denote the midpoint of arc 
.
meets 
.
at 
and 
meets 
a second time at 
,
meets 
a second time at 
.
passes through the antipodal to 
intersects the lines 
,
at points 
,
,
of the segments 
,
,
,
.
be a positive integer. Prove that there are infinitely many primes, each of which divides at least one term of the sequence 
,
.
Y by
A triangle is either isosceles (which for the purpose of this question we shall take to include equilateral) or scalene (all sides different).
(a) How many triangles are there with integer-length sides and perimeter
?
(b) How many triangles are there with integer-length sides and perimeter
?
(c) How many isosceles triangles are there with integer-length sides and perimeter
(where 
is an integer) ?
(d) How many triangles are there altogether with integer-length sides and perimeter (where is an integer) ?
(a) How many triangles are there with integer-length sides and perimeter
?(b) How many triangles are there with integer-length sides and perimeter
?(c) How many isosceles triangles are there with integer-length sides and perimeter
(where 
is an integer) ?(d) How many triangles are there altogether with integer-length sides and perimeter
(where is an integer) ?
.
shall be less than half of the perimeter and not less than one third of the perimeter, i.e. 
.
,
can be 
.
such triangles;
,
.
such triangles;
,
.
such triangles;
,
.
such triangles.
triangles with integer-length sides and perimeter of 
.
.
triangles with integer-length sides and perimeter of 
.
.
,
.
such triangles.
,
.
such triangles.
isosceles (including equilateral) triangles with integer-length sides and perimeter of 
.
,
