Mathcounts Chapter Competition Solutions Worksheet - Middle School - 2014 Page 11
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12that cost the same with either plan.
Cromwell chose the last prime. But we want
Let x represent the number of shirts for which
Bartholomew to end up with the last prime.
the cost is the same using either plan. Now,
So let's suppose, instead, that the game
we solve 250 + 4.25x = 150 + 5.25x and get
proceeds this way:
x = 100 Ans.
A chooses 2
B adds 3 to make 5
C adds 2 to make 7
5. $1000 is in an account that earns 3%
A chooses 4 to make 11
interest, compounded annually.
B chooses 2 to make 13
The account now is worth $1125.51. How
C chooses 4 to make 17
many years has the money been in the
A chooses 2 to make 19
account? The table shows the year-end
B chooses 4 to make 23
balance of an account with an initial balance
This time, Bartholomew has chosen the last
of $1000 that earns 3% interest,
prime, and the total number of primes made
compounded annually.
was 8 Ans.
Year
Ending Balance
7. How many positive integers less than 1000
1
1.03 × 1000 = 1030
do not have 7 as any digit?
2
1.03 × 1030 = 1060.90
3
1.03 × 1060.90 = 1092.727
Let’s start with positive 3-digit integers.
4
1.03 × 1092.73 = 1125.50881
The possible values for the digit the hundreds
place are 1 through 6, 8 or 9. That’s 8
That means, 4 years after the account balance
choices. The possible values for the digit the
was $1000, the account earned enough
tens and ones places are of 0 through 6, 8 or
interest to have a balance of $1125.51.
9. That’s 9 choices for the tens place value
4 Ans.
and 9 choices for the ones place value. Thus,
there are 8 × 9 × 9 = 648 positive 3-digit
integers. For positive 2-digit integers, the
6. Abigail, Bartholomew and Cromwell play a
possible values for the digit in the tens place
game in which they take turns adding 1, 2, 3,
are 1 through 6, 8 or 9. That’s 8 values. And
or 4 to a sum in order to create an increasing
again, there are 9 choices for the ones place
sequence of primes.
value. So there are 8 × 9 = 72 positive
We'll let A, B and C stand for Abigail,
2digit integers. Finally, the positive 1-digit
Cromwell and Bartholomew, respectively.
integers are 1, 2, 3, 4, 5, 6, 8 and 9. That’s
Let's suppose the game proceeds as follows:
8 positive 1-digit integers. That brings the
A chooses 2
total number of integers to 648 + 72 + 8 =
B adds 1 to make 3
728 Ans.
C adds 2 to make 5
A chooses 2 to make 7
B chooses 4 to make 11
8. In the game, vowels are worth 1 point and
C chooses 2 to make 13
consonants are worth 2 points. When more
A chooses 4 to make 17
than one letter of the same type appears
B chooses 2 to make 19
consecutively, each letter is worth twice as
C chooses 4 to make 23
much as the one before. Find the absolute
A chooses...
difference between the values of QUEUEING
Well, A can’t choose, since there are no
and SYZYGY. Following are the point values
primes between 24 and 27. That means
for each letter in QUEUEING.
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