:
such that:
for all x,y E 
,
,
,
be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, 
.
for all 
.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
be a trapezoid inscribed in circle 
,
.
is the symmetric point of 
across 
,
intersects 
.
is orthocenter of triangle 
.
take a point 
so that 
.
coincide
Product of all even divisors of 
If 
dice are rolled, Then number of ways of getting sum at least 
is
foot radius. Bruno completes 
laps every hour, while Brutus completes 
laps every hour. If they start at the same point but run in opposite directions, how far along the track’s circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point.
,
,
,
such that ![\[\left(a_1+\frac1{a_1}\right)\left(a_2+\frac1{a_2}\right)\cdots\left(a_n+\frac1{a_n}\right)=2020.\]](http://latex.artofproblemsolving.com/3/6/2/362fccfa6353ffcbc4d29f24ccbfce73269080bf.png)
Compute 
.
.
side.
at point 
at 
.
A
have angles 
and 
such that 
and 
. Moreover, suppose that the product of the side lengths of the triangle is equal to its area. Let 
denote the circumcircle of 
intersect 
at 
intersect 
at 
,
intersect 
at 
can be written as 
for relatively prime integers 
and 
and squarefree 
,
is 
)
divisors.
.
such that 
is a positive integral power of a prime positive integer."
Y by Adventure10, Mango247
Suppose 
is a Simson line of the triangle 
that passes through 
.
Suppose
is a cyclic hexagon such that 
intersect at a single point. Prove that 
is a rectangle.
Should the first sentence read:
Suppose is a Simson line of the triangle with respect to .
? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.
is a Simson line of the triangle 
that passes through 
.Suppose
is a cyclic hexagon such that 
intersect at a single point. Prove that 
is a rectangle.Should the first sentence read:
Suppose
is a Simson line of the triangle with respect to .? Since it appears weird that a Simson line that passes a point is to be constructed. However, this is Unsolved after all, so I'm not sure.
