Well, I have a challenge for you people.
1.
In this problem, a half deck of cards consists of 26 cards, each labled with an integer from 1 to 13. There are two cards labled 1, two labled 2, two labled 3 etc. A certain math class has 13 students. Each day, the teacher thoroughly shuffles a half deck of cards and deals out two to each student. Each student then adds the two numbers on the cards recieved, and the resulting 13 sums are multiplied together to form a product P. If P is an evennumber, then the class must do math homework that evening. Show the the class must always do math homework.
2.
26 people attended a math party: Archemides, Bernouli, Cauchi, ..., Yau, Zeno. During the party, Archemides shook hands with 1 person, Bernoili shook hand with 2 people , Cauchy shook hands with 3 people, and similarly up to Yau who shook hands with 25 people. How many people did Zeno shake hands with? Justify that you answer is correct and that it is the only correct answer.
3.
Prove tyhat there are no integers m,n >,=1 such that
sqrt(m+sqrt(m+sqrt(m+ ... +sqrt(m)=n
where there are 2006 square root signs.
Much obliged if someone can make a picture for this.
Good luck.
