Solutions to Set 3

• Problem #1: 10
• Problem #2: medium
• Problem #3: 1/2
• Problem #4: 7
• Problem #5: 6
• Problem #6: 6006
• Problem #7: 3
• Problem #8: 54
• Problem #9: 1
• Problem #10: 270

Explanations

• Problem #1: There is a large right triangle with hypotenuse 17. The total length of its base is 9+6=15. Using the Pythagorean theorem or knowledge of the Pythagorean triple (8, 15, 17), you can then conclude that the height of the triangle is 8.

  There is also a smaller right triangle with hypotenuse AB. Its base, 6, is given, and you have just figured out that its height is 8. Using the Pythagorean theorem or knowledge of the Pythagorean triple (6, 8, 10) (a multiple of (3, 4, 5)), you can then conclude that the length of AB is 10.

  Read more about The Pythagorean Theorem.

• Problem #2: The large pizza has an area of 81π sq. in. The medium pizza has an area of 64π sq. in. The small pizza has an area of 36π sq. in.

  Therefore, the large pizza provides 81π/15 = 5.4π sq. in. per dollar. The medium pizza provides 64π/10 = 6.4π sq. in. per dollar. The small pizza provides 36π/7 ≈ 5.1π sq. in. per dollar.

  Since the medium pizza provides the greatest area per dollar, medium is the best deal.

  Read more about The Best Deal.

• Problem #3: Let S represent the sum of the 25 numbers. By definition, the arithmetic mean is S/25. Since the arithmetic mean is 8% of N, you can write the equation
  S/25 = .08N.

  Multiply both sides by 25 and you will get
  S = 2N
  S/N = 2
  N/S = 1/2

  Read more about The Arithmetic Mean in Equations.

• Problem #4: The original number leaves a remainder of 4 when divided by 9, so it is 4 more than a multiple of 9 and can be written as 9n+4.

  When you multiply that expression by 4, you get 36n+16.

  This leaves a remainder of 7 when divided by 9. Algebraically, this is true because
  36n + 16
  = 36n + 9 + 7
  = 9(4n + 1) + 7
  so this number is 7 more than a multiple of 9.

• Problem #5: There are two key things that you should notice. First of all, just determining the first three letters of the palindrome will determine the whole thing. Second of all, due to the symmetry of palindromes, those first three letters must include exactly 1 A, 1 H, and 1 N.

  Therefore, you need to count the number of 3-letter permutations of A, H, and N. The number of permutations of 3 things chosen 3 at a time is 6.

  The 6 permutations are:
  AHNNHA
  ANHHNA
  HANNAH
  HNAANH
  NAHHAN
  NHAAHN

  Read more about Calculating Permutations.

• Problem #6: The number of diagonals in a 2005-sided polygon is (2005*2002)/2, and the number of diagonals in a 2002-sided polygon is (2002*1999)/2. Therefore,

  f(2005) - f(2002)
  = (2005*2002)/2 - (2002*1999)/2
  = 2005*1001 - 1999*1001
  = 1001(2005 - 1999)
  = 1001(6)
  = 6006.

  Read more about Formulas About Polygons.

• Problem #7: Let a, b, and c represents the number of chickens, airplanes, and cows, respectively. From the given information, three algebraic equations can be written:

  Eyes: 2a + 2c = 44
  Wings: 2a + 2b = 30
  Legs: 2a + 4c = 64

  Subtract the first equation from the third, and you will get:

  2c = 20
  c = 10.

  Continue to solve for a, and then solve for b. You should end up with
  a = 12
  b = 3.

  Therefore, there are 3 airplanes (along with 12 chickens and 10 cows).

  Read more about Translating Word Problems.

• Problem #8: Let r represent the radius of the sphere in yards, and let 3r represent the radius of the sphere in feet. Use the formulas for the volume and surface area of a sphere to set up an algebraic equation:

  (4/3)πr³ = 4π(3r)²
  (4/3)πr³ = 4π(9r²)
  (4/3)r³ = 4(9r²)
  r³ = 3(9r²)
  r³ = 27r²)
  r = 27.

  Since the radius of the sphere is 27 yards, the diameter is 54 yards.

  Read more about Formulas About Spheres.

• Problem #9: If Fanny is a Circle, then Anna-Marie-Juniper's statement is true, making Anna-Marie-Juniper a Circle as well. If Fanny is a Square, then Anna-Marie-Juniper's statement is false, making Anna-Marie-Juniper a Square as well. Either way, Fanny and Anna-Marie-Juniper are of the same nationality.

  Since Fanny and Anna-Marie-Juniper are of the same nationality, Gonzo's statement is true. Therefore, Gonzo is a Circle. This means that Fanny's statement is false, making Fanny a Square. Since Fanny is a Square, Anna-Marie-Juniper is a Square as well.

  Therefore, Gonzo is a Circle, while Fanny and Anna-Marie-Juniper are Squares. There is only one Circle.

• Problem #10: By definition,
  10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1.

  So, the prime factorization of that number is:
  10! = (5 * 2) * 3² * 2³ * 7 * (2 * 3) * 5 * 2² * 3 * 2
  = 7 * 5² * 3⁴ * 2⁸.

  Using the formula for counting factors, 10! therefore has
  (1 + 1) * (2 + 1) * (4 + 1) * (8 + 1)
  = 2 * 3 * 5 * 9
  = 270 factors.

  Read more about Counting Factors.