Challenging Problems of October

  1. If a, b and c are three consecutive integers, which of the following must be true?

    2. I. At least one of these integers is divisible by 2.

    II. At least one of these integers is divisible by 3.

    III. One of these integers is divisible by 4.

  2. The degree measures of the angles of a triangle are x, y and z. If x, y and z are integers and x < y < z, what is the least possible value of z?
  3. In the figure shown, the circles with centers O and R each have a radius of 2. If PQ = 1, then what is the perimeter of the rectangle KLMN?

  4. I wish to share thirty identical individual sausages equally among eighteen people. What is the minimum number of cuts that I need to make?
  5. In a certain village live 800 women. Three percent of them are wearing one earring. Of the other 97 percent, half are wearing two earrings and half are wearing none. What is the total number of earrings being worn by the women?
  6. A woman has two grandfather clocks in her house. One of the clocks does not run at all, and the other clock always loses an hour a day. Which of the clocks will display the correct time most often during any given week?
  7. Jessica, Amy, Marybeth and Tracie all eat breakfast together every morning. Each morning they like to line up differently. How many days can they go without repeating the same order?
  8. On Dick’s birthday, his brother Harry is seventeen years younger than three times Dick’s age. The boys' father, Tom, is twelve years older than twice Harry’s age. If Dick is seven years younger than his brother, how many candles are on Dick’s cake?
  9. What is the tens digit of 0! + 1! + 2! + 3! + … + 2000!? (Note: 0! = 1, 1! = 1, 2! = 1x2, 3! = 1x2x3, 4! = 1x2x3x4, …)
  10. In 1930, a correspondent proposed the following question: A man’s age at death was one twenty-ninth of the year of his birth. How old was the man in 1900?
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