,
.
be an equilateral triangle.M and N are the midpoints of 
and 
respectively. Externally to the triangle 
.
is the intersection of 
and 
,
is a bisector of the angle 
bears — Aruno, Bruno, Cruno and Druno — are each given a card with a positive integer and are told that the sum of their 
They cannot show each other their cards, but discuss a series of observations in the following order:
by 
Compute all complex numbers 
such that 
such that 
different positive integers.
.
be a dot on the 
side.
at point 
at 
.
A
and 
have equal side length 
They intersect at 
and 
Given that 
and 
what is 
?
denote the number of positive integers less tha
denote the set of positive integers 
is expressed as a simplified fraction, the denominator is a power of 
.
that satisfies the following:
is not divisible by any square of a positive integer greater than 
,
be a degree 
polynomial satisfying 
for 
Find 
.
be the midpoint of 
.
,
in degrees.
be an isosceles triangle with 
Let 
Let 
intersect the extension of 
at 
Let 
be the midpoint of segment 
and let 
be the intersection of 
and 
Let the extension of 
intersect 
and the circumcircle of 
and 
respectively. Given that 
and 
compute the length of 
be a circle, and let a line 
intersect 
Circles 
and 
are internally tangent to 
and 
respectively, and both are tangent to 
Similarly, circles 
and 
are externally tangent to 
and 
respectively.
the segment 
has a length of 
and 
find the length of segment 
-
-
given that the repetend has a length of 
?
Y by
On the plane through the same distance are 
parallel lines. Each line has one point. Points 
and 
are taken arbitrarily on the first two lines. Then point 
is taken so that 
so that 
, ..., 
so that 
,..., 
so that 
. Moreover, if the next point can be selected in two ways, then for the odd number the right point is selected, for the even - the more left point (see figure). Prove that the location of point depends only on the location of point .
parallel lines. Each line has one point. Points 
and 
are taken arbitrarily on the first two lines. Then point 
is taken so that 
so that 
, ..., 
so that 
,..., 
so that 
. Moreover, if the next point can be selected in two ways, then for the odd number the right point is selected, for the even - the more left point (see figure). Prove that the location of point depends only on the location of point .This post has been edited 2 times. Last edited by parmenides51, Jul 4, 2020, 6:29 PM
