15
Part of 1985 ITAMO
Problems(2)
Tournament Points
Source:
3/14/2006
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten). What was the total number of players in the tournament?
combinatoricscombinatorial game
Square to Polyhedron
Source:
3/15/2006
Three 12 cm 12 cm squares are each cut into two pieces and , as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in ) of this polyhedron?[asy]
defaultpen(fontsize(10));
size(250);
draw(shift(0, sqrt(3)+1)*scale(2)*rotate(45)*polygon(4));
draw(shift(-sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(165)*polygon(4));
draw(shift(sqrt(3)*(sqrt(3)+1)/2, -(sqrt(3)+1)/2)*scale(2)*rotate(285)*polygon(4));
filldraw(scale(2)*polygon(6), white, black);
pair X=(2,0)+sqrt(2)*dir(75), Y=(-2,0)+sqrt(2)*dir(105), Z=(2*dir(300))+sqrt(2)*dir(225);
pair[] roots={2*dir(0), 2*dir(60), 2*dir(120), 2*dir(180), 2*dir(240), 2*dir(300)};
draw(roots[0]--X--roots[1]);
label("", centroid(roots[0],X,roots[1]));
draw(roots[2]--Y--roots[3]);
label("", centroid(roots[2],Y,roots[3]));
draw(roots[4]--Z--roots[5]);
label("", centroid(roots[4],Z,roots[5]));
label("", (1+sqrt(3))*dir(90));
label("", (1+sqrt(3))*dir(210));
label("", (1+sqrt(3))*dir(330));
draw(shift(-10,0)*scale(2)*polygon(4));
draw((sqrt(2)-10,0)--(-10,sqrt(2)));
label("", (-10,0));
label("", centroid((sqrt(2)-10,0),(-10,sqrt(2)),(sqrt(2)-10, sqrt(2))));[/asy]
geometry3D geometrytetrahedron