be 
.
meet at point 
.
on line 
which is not on the segment 
be 
.
meet 
respectively. Let 
be the midpoint of 
lie on a circle.
red cards, 
white cards, and 
blue cards. A player plays all the cards one at a time. With each play he accumulates a penalty. If he plays a blue card, then he is charged a penalty which is the number of white cards still in his hand. If he plays a white card, then he is charged a penalty which is twice the number of red cards still in his hand. If he plays a red card, then he is charged a penalty which is three times the number of blue cards still in his hand. Find, as a function of 
and 
the minimal total penalty a player can amass and all the ways in which this minimum can be achieved.
and a set 
consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
,
in at least 
different ways.
lies on side 
lies on side 
.
be points on segments 
and 
,
is parallel to 
.
be a point on line 
,
.
be the point on line 
,
.
,
Y by Adventure10
Let 
be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point 
. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral bisects the lines 
(
is the center of the circle circumscribed around quadrilateral ).
(Alexander Dunyak)
be an inscribed quadrilateral whose diagonals are connected internally. are perpendicular to each other and intersect at the point 
. Prove that the line connecting the midpoints of the opposite sides of the quadrilateral bisects the lines 
(
is the center of the circle circumscribed around quadrilateral ).(Alexander Dunyak)
This post has been edited 1 time. Last edited by parmenides51, Oct 11, 2020, 10:39 PM
