Math Guru: HS Math Competition Question 1 - 珍珠湾ART

Question 1: Space aliens decide to build another ring around Saturn. They draw a circle and place two rocks, each with mass 1 gram, that cut the circle into two half circles. Then the first alient looks at each half circle and puts a rock with mass 2 grams in the middle of each of the two arcs. The circle is thus devided into 4 quarter circles. For each of the four resulting arcs, the second alient adds up the masses of the rocks at the two ends of the arc and places a rock with that mass in the middle of the arc. The third alient then looks at each of the resulting 8 arcs, adds up the masses of the rocks that are at the two ends of the arc, and places a rock with that mass in the middle of the arc. They continue in this fashion, with each successive alient adding up the masses at the two ends of each arc and dividing the arc in two by placing a rock, with mass equal to this sum, in the middle of the arc. They stop as soon as the total mass of the rocks they have placed is greater than the mass of Saturn, namely 6x10^29 grams. Let n be the number of aliens who placed rocks on the circle. Which of the following is true?

a. 1

b. 20

c. 100www.ddhw.com

d. 500

e. 2005

Question 2: There are 2005 people at a party. By the end of the party, en every group of four people there is at least one person who has shaken hands with the other three members of the group. What is the smallest possible number of people who have shaken hands with everyone at the party?

Question 3: A group of N islands are connected by bridges. Each island has bridges to at most 3 other islands. One can travel between any 2 islands by crossing at most two bridges. What is the largest possible value of N? (Bridges are allowed to go over or under other bridges.)

Question 4: Whenever trick-or-treaters come to her door on Halloween, a mathematician makes them choose two positive real numbers x and y. She lets s be the smallest of x, y+1/x, and 1/y and gives them s pounds of candy. What is the largest possible value of s?

Question 5: After all twenty participants in a figure skating tournament skated, each of the 9 judges ordered the participants from place 1 (the best) to palce 20 (the worst). It turned out that for each participant, the places assigned by different judges were not more than 3 apart. The sum of the places for each participant was calculated and the sums were ordered:

c1 <= c2 <= ... <= c20. What is the largest possible value of c1?