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BMAD

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  1. An odd number of soldiers are stationed in a field such that all the pairwise distances are distinct. Each soldier is told to keep an eye on the nearest other soldier. Prove that at least one soldier is not being watched.
  2. Each person is betting $10 that they can pick the lowest positive integer that is not picked by anyone else. Each time an individual chooses a number they bet $10. Once everyone is satisfied that they picked enough numbers (as they can pick more than one) they show their choices. The individual with the lowest number that was not picked by anyone else, wins $100. You are competing against nine other logical and equally wealthy people, what would be your strategy to win the prize?
  3. how are we defining touching?
  4. Suppose you have a point within a equilateral triangle. If you were to connect each vertex to this point you would make three new line segments. Assume that you knew two of the angles formed at the point. Build a triangle out of these line segments. What can the two known angles tell us about the angles of this newly created triangle.
  5. Stefan was on his way out the door to visit an old friend across the village when he realized that his grandfather clock had stopped and no longer displayed the correct time. This was the only clock in his home and the man owned no watches or other time-telling devices. Without disappointment Stefan left his home and walked roughly three miles to his friend's house. He glanced at the friend's wall clock as he entered the house and after visiting for a few hours set off back home along the same route. He walked at the same pace home and had no idea of knowing how long his trip back took him
  6. There was a flock of ducks flying in the sky. One ahead of the other two; one behind of the other two; one between the other two, and three in a row. How many ducks were flying?
  7. I know what you mean but I don't think it is worded it right
  8. Bill sold his motor scooter to Tom for $100. After driving it around for a few days Tom discovered it was in such a broken-down condition that he sold it back to Bill for $80. The next day Bill sold it to Herman for $90. What is Bill's total profit?
  9. You and two of your friends would like to know the average of all of your salaries. You are each self-conscious about the amount of money you make and will not tell one another your salaries. What can you do to figure out the average salary?
  10. You are in an empty room with a glass of water. The glass is a right cylinder that looks like it is about half-full, but you are unsure. What is the most accurate way, without spilling any water, to determine whether the glass is half-full, more than half-full, or less than half-full?
  11. A man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit either girl he must take a train. To go to the one in Brooklyn, he takes a train on the downtown side of the platform, and to visit the one in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally well, he simply takes the first train that comes along. In this way he lets chance determine whether he rides to the Bronx or to Brooklyn. The young man reaches the subway platform at a random moment each Saturday afternoon.
  12. There is a solution. the problem can be solved by them either being parallel or not intersecting. Since this is proving challenging, lets just focus on the parallel situation for now. Maybe I will repost the question for not intersecting later.
  13. A game token costs $10 to play. The pay out is $100. You can purchase multiple entries if you desire. For each entry you purchase, you must pick the lowest positive number that no one else picks. If there are ten people, including yourself, seeking to purchase tokens, what is your strategy?
  14. Prove that x^(1/x) = x^-x has only one solution
  15. You and four friends are playing Russian roulette, one bullet is in a chamber of a six chamber gun. Each of you must take a shot from the gun. The chamber will only be spun once, before anyone has taken a shot. If you got to chose, which position, 1st, 2nd, 3rd, 4th, or 5th would best help your chances of survival?
  16. In front of you is a line and a line segment. You need to dissect the line segment into 6 equal segments. Unfortunately you only possess two skills. you have the ability to mark points and you have the ability to draw straight segments. How will you segment the segment into six equal parts?
  17. Here is a list of months and a code for each January: 7110 February: 826 March: 5313 April: 541 May: 3513 June: 4610 July: 4710 What is the code for the month of August?
  18. For 0<x<y find an integer solution for x^y = y^x
  19. A number's persistence is the number of steps required to reduce it to a single digit by multiplying all its digits to obtain a second number, then multiplying all the digits of that number to obtain a third number, and so on until a one-digit number is obtained. For example, 77 has a persistence of four because it requires four steps to reduce it to one digit: 77-49-36-18-8. The smallest number of persistence one is 10, the smallest of persistence two is 25, the smallest of persistence three is 39, and the smaller of persistence four is 77. What is the smallest number of persistence five?
  20. Suppose superman can survive the vacuum of space (the comics are inconsistent with this ability). Let's say he is in the front of a spaceship that just crossed, what I believe in English is called, the event horizon for a black hole. That point where matter gets sucked into hole. Is it possible for superman to move to the rear of the ship and escape the black hole, assuming that the back of the ship hasn't crossed the event horizon?
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