41. A fifty feet rope is tied on either end to a goat and a post. The post is located in the middle of two fences fifty feet apart. Assuming the goat can't chew threw the rope or jump the fence, how many square feet is the goat able to roam? (1 point)

42. Find x. Report your exact answer without any trig functions. (2 points)

43. Eight pennies are arranged as shown in the picture. If the top penny , fully pictured, is rolled without slipping, around the other seven, how, many complete somersaults will Mr. Lincoln make as the penny comes back to it’s original position? (1 point)

44. At an international math conference, the badge also printed the languages spoken by the mathematicians. At a table of 7 mathematicians, one noted that each in the group speaks at most two languages and among any three there are at least two who can communicate. Prove that there are three mathematicians speak a common language (a language other than math!) (2 points)

45. Solve for x, (1 point).

46. Each holiday season, six friends toss their names into a hat and vigorously mix the names. They take turns drawing the names out of the hat with the understanding that they are to buy a small gift for the selected individual. If anyone selects her own name, the whole affair is adjourned until next year. What is the probability that gifts are exchanged on any given year? (2 Points)

47. A peasant declines a reward of gold and jewels. Instead he prefers to get his reward in rice over a sixty-four day period: a single grain on the first day, two grains on the second day, four grains of rice on the third day, and so on, each day receiving twice as much rice as on the previous day. Exactly how many grains of rice is the peasant's reward? (2 Points)

48. A cone is made by rolling a sheet of plastic such that the the opening forms a two foot diameter circle. The tip of the cone forms a right angle when viewed edge on. What is the area of the outside surface of this cone? (Do not include the area of the opening.) (2 Points)

49. Two identical raw (straight) spaghetti noodles can be positioned so that they make a cross. No matter how three noodles are positioned, they will cross no more than three times. What is the maximum number of times 10 noodles can cross each other? ( 2 Points)

50. Memorize the powers of 2 up to the twelfth power. Be able to answer questions like "what is two to the eighth?" without having to count up from 2, 4, ... (1 Point)

45. Solve for x, (1 point).

46. Each holiday season, six friends toss their names into a hat and vigorously mix the names. They take turns drawing the names out of the hat with the understanding that they are to buy a small gift for the selected individual. If anyone selects her own name, the whole affair is adjourned until next year. What is the probability that gifts are exchanged on any given year? (2 Points)

47. A peasant declines a reward of gold and jewels. Instead he prefers to get his reward in rice over a sixty-four day period: a single grain on the first day, two grains on the second day, four grains of rice on the third day, and so on, each day receiving twice as much rice as on the previous day. Exactly how many grains of rice is the peasant's reward? (2 Points)

48. A cone is made by rolling a sheet of plastic such that the the opening forms a two foot diameter circle. The tip of the cone forms a right angle when viewed edge on. What is the area of the outside surface of this cone? (Do not include the area of the opening.) (2 Points)

49. Two identical raw (straight) spaghetti noodles can be positioned so that they make a cross. No matter how three noodles are positioned, they will cross no more than three times. What is the maximum number of times 10 noodles can cross each other? ( 2 Points)

50. Memorize the powers of 2 up to the twelfth power. Be able to answer questions like "what is two to the eighth?" without having to count up from 2, 4, ... (1 Point)