Download Level 2: Part B The 25th Math Bowl March 6, 2010 1. Anne, Bob

The 25th
Level 2: Part B
Math Bowl
March 6, 2010
Write your answers in the Blue Book. Print your names & write the number of students taking this test in
the upper right corner of the Blue Book. Put this test & the Blue Book in the provided envelope.
For credit and partial credit show your work.
1. Anne, Bob, Carol, and Don are rescued from a desert island by a pirate who forces them to play a game.
Each of the four, in alphabetical order by first name, has to roll two dice. If the total on the dice is
either 8 or 9, the person rolling the dice has to walk the plank. The game stops as soon as one player
loses or after all have rolled the dice once. What is the probability that Don survives?
Solution: Don does not survive if Anne, Bob, and Carol all survive, but he loses. The chance that
any individual player loses on his or her turn is 41 , since there are nine ways to roll an 8 or a 9, out of
27
, and the chance
36 possible rolls. Therefore the chance that Don does not survive is 43 · 34 · 43 · 14 = 256
27
229
that he survives is 1 − 256 = 256 .
2. Find all three-digit numbers with the property that the sum of the three digits is equal to the product
of the three digits.
Solution: Let the three-digit number be abc = a · 100 + b · 10 + c, where 1 ≤ a ≤ 9, 0 ≤ b ≤ 9, 0 ≤ c ≤ 9.
We want a + b + c = a · b · c. Since a 6= 0, we see that also b 6= 0, c 6= 0, since otherwise the product
would be 0 but the sum is not. Let k be the largest of the three numbers; then the sum of the three
numbers is at most 3k, which means that the product of the remaining two numbers is at most 3, so
the remaining two numbers are 1 and 1, 1 and 2, or 1 and 3. In the first case, the sum of the numbers
is then k + 2, so we would have k + 2 = k · 1 · 1, or 2 = 0, an impossibility. In the second case, the sum
of the numbers is then k + 3, so we would have k + 3 = 2k, so {a, b, c} = {1, 2, 3} is a possibility. In the
third case, k + 4 = 3k, so 4 = 2k and k = 2, contradicting the assumption that k is the largest of the
three numbers. Thus the set of three-digit numbers with this property is {123, 132, 213, 231, 312, 321}.
3. If a +
1
a
= 5, find a2 +
1
a2
and a4 +
1
a4 .
1
1
1
1
1
Solution: (a + )2 = a2 + 2 + 2 = 25, so a2 + 2 = 25 − 2 = 23. Then (a2 + 2 )2 = a4 + 4 + 2 =
a
a
a
a
a
1
232 = 529, so a4 + 4 = 529 − 2 = 527.
a
4. Four distinct positive integers a, b, c, d form an arithmetic sequence1 . The integers a, b, d also form a
geometric sequence.2 The sum of the four integers is a perfect cube. Find the least possible value for a,
and describe all possible values for a.
Solution: Let k be the common difference for the arithmetic sequence and let r be the common ratio for
the geometric sequence. Then r = ab = db , so b2 = da. But b = a + k, d = a + 3k, so (a + k)2 = (a + 3k)a.
Then a2 + 2ka + k 2 = a2 + 3ka, giving ka = k 2 , so a = k. Then b = a + k = 2a = ar so r = 2. The
sum of the four integers is then a + 2a + 3a + 4a = 10a = n3 for some integer n. Then we must have
a = 102 m3 for some integer m, and any such a will satisfy the given condition. The least possible value
occurs when m = 1 and a = 100.
1 Each
2 Each
term is formed by adding a constant to the proceeding term.
term is formed by multiplying a constant times the proceeding term.
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Please continue
Math Bowl
The 25th
Level 2: Part B
March 6, 2010
5. The horizontal rectangle below has length a and height b. Find the area and perimeter of the slanted
rectangle.
a
b
Solution: Draw a perpendicular from the diagonal of the horizontal rectangle to the lower right corner
of the horizontal rectangle. This divides the slanted rectangle into two smaller rectangles, whose
diagonals are formed by the sides of the first rectangle. This makes it clear that the triangle formed by
the diagonal of the first rectangle and these two sides make up exactly half of the slanted rectangle, so
that its area is exactly equal to that of the first rectangle, or a · b square units. To find the perimeter,
√
let h be the length of the shorter side of the slanted rectangle. The longer side has length a2 + b2 ,
p
√
2ab
=
, and the perimeter of this square is 2 a2 + b2 + √
and h · a2 + b2 = a · b, so h = √aa·b
2 +b2
a2 + b 2
2(a2 + b2 + ab)
√
.
a2 + b 2
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-End of Level 2: Part B-