consecutive even numbers, such that the square of their product is divisible by the sum of their squares.
points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into 
pairs and cover each pair of points with a circle containing no other points.
is called pretty if it has finite expression in bas
is a positive integer in ![$[2;2022]$](http://latex.artofproblemsolving.com/f/0/5/f05e359b3b158e7a4a742cbe662e8d1417fd7926.png)
.
that with every 
in 
then there is at least one pretty number between 
and 
and 
.
.
be a triangle and let 
be the midpoint 
.
such that 
and 
. Prove that line 
passes through the midpoint of segment 
.
,
and 
are all irrational numbers.
and a minimum value of 
over the interval ![$[-1, 1]$](http://latex.artofproblemsolving.com/5/c/3/5c3818b9565a33fd3aadba10026d32c5e3eea90f.png)
?
Prove that
exist that satisfy 
for all real 
,
?
and 
Prove that
Let 
Prove that
.
?
Y by
Dear Mathlinkers,
1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. X the point of intersection of the parallel to AB through Q and BC
7. Y the point of intersection of the segment AX and (A)
Prove : CY is parallel to IP.
Jean-Louis
1. ABCD a square
2. (A) the circle with center at A passing through B
3. P the points of intersection of the segment AC and (A)
4. I the midpoint of AB
5. Q the point of intersection of the segment IC and (A)
6. X the point of intersection of the parallel to AB through Q and BC
7. Y the point of intersection of the segment AX and (A)
Prove : CY is parallel to IP.
Jean-Louis
be 
,
be 
,
be 
.
,
,
,
,
.
.
,
,
,
is parallel to 
