Download Sprint Round

2006 Mathcounts Training, Sprint Round
1. Express
22001 · 32003
as reduced fraction.
62002
2. For the nonzero numbers a, b, and c, define
✤(a, b, c) =
abc
.
a+b+c
Find ✤(2, 4, 6).
3. The arithmetic mean of the nine numbers in the set
{9,99,999,9999, . . . ,999999999}
is a 9-digit number M , all of whose digits are distinct. Which digit
does not appear in M ?
4. What is the value of
(3x − 2)(4x + 1) − (3x − 2)4x + 1
when x = 4?
5. Circles of radius 2 and 3 are externally tangent and are circumscribed
by a third circle, as shown in the figure. Find the area of the shaded
region in terms of π.
6. Express
2 − 4 + 6 − 8 + 10 − 12 + 14
3 − 6 + 9 − 12 + 15 − 18 + 21
as a reduced fraction.
7. Al gets the disease algebritis and must take one green pill and one pink
pill each day for two weeks. A green pill costs $1 more than a pink pill,
and Al’s pills cost a total of $546 for the two weeks. How much does
one green pill cost?
8. The sum of 5 consecutive even integers is 4 less than the sum of the
first 8 consecutive odd counting numbers. What is the smallest of the
even integers?
9. For how many integers n is
n
20−n
the square of an integer?
10. Each row of the Misty Moon Amphitheater has 33 seats. Rows 12
through 22 are reserved for a youth club. How many seats are reserved
for this club?
11. Rose fills each of the rectangular regions of her rectangular flower bed
with a different type of flower. The lengths, in feet, of the rectangular
regions in her flower bed are as shown in the figure. She plants one
flower per square foot in each region. Asters cost $1 each, begonias
$1.50 each, cannas $2 each, dahlias $2.50 each, and Easter lilies $3
each. What is the least possible cost, in dollars, for her garden?
12. How many two-digit positive integers have at least one 7 as a digit?
13. At each basketball practice last week, Jenny made twice as many free
throws as she made at the previous practice. At her fifth practice she
made 48 free throws. How many free throws did she make at the first
practice?
14. A standard six-sided die is rolled, and P is the product of the five
numbers that are visible. What is the largest number that is certain to
divide P ?
15. The digits 0, 1, 2, and 3 are used to replace a, b, c and d in the expression
c · ab − d. What is the maximum possible value of the result?
16. A scout troop buys 1000 candy bars at a price of five for $2. They sell
all the candy bars at a price of two for $1. What was their profit, in
dollars?
17. A positive number x has the property that x% of x is 4. What is x?
18. A gallon of paint is used to paint a room. One third of the paint is used
on the first day. On the second day, one third of the remaining paint
is used. What fraction of the original amount of paint is available to
use on the third day?
19. For real numbers a and b, define a b =
√
a2 + b2 . What is the value of
(5 12) ((−12) (−5)) ?
Express your answer in simplest radical form.
20. Brianna is using part of the money she earned on her weekend job to
buy several equally-priced CDs. She used one fifth of her money to buy
one third of the CDs. What fraction of her money will she have left
after she buys all the CDs?
21. What is the smallest positive integer n such that n!(n + 1)!(n + 2)! is
a perfect square?
22. If a and b are digits for which
2
×b
6
9 2
9 8
a
3
9
9
then a + b =
23. The adjacent sides of the decagon shown meet ..at
right angles.
What is
..
•.. ............................•
...
.
..
2.
..
•....................•.
..
................•..
•
...
..
..
..
...
..
..
8 ....
..
..
..
..
...
•.........................•..
..
..
its perimeter?
•............................................................................•
12
24. If x, y, and z are real numbers such that
(x − 3)2 + (y − 4)2 + (z − 5)2 = 0,
then x + y + z =
25. If a is 50% larger than c, and b is 25% larger than c, then a is what
percent larger than b?
26. A rectangle with perimeter 176 is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five
congruent rectangles?
27. Consider the sequence
1, −2, 3, −4, 5, −6, . . . ,
whose nth term is (−1)n+1·n. What is the average of the first 200 terms
of the sequence? Express your answer in decimal notation, rounded to
the nearest tenth.
28. The sum of seven integers is −1. What is the maximum number of the
seven integers that can be larger than 13?
29. The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x?
..
..
D ... E
....................
..
B ... C
....................
.
x ... A
..
30. How many base 10 four-digit numbers,
N = a b c d, satisfy all three of
.
the following conditions?
(i ) 4, 000 ≤ N < 6, 000; (ii ) N is a multiple of 5; (iii ) 3 ≤ b < c ≤ 6.