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Warm-Up 7
1. ________ A triangle with a height of 24 inches has the same area as a rectangle 12 inches by
6 inches. How many inches long is the base of the triangle that corresponds to the
24-inch height?
2. ________ Rohan plans to open a bank for his friends. Some friends will allow him to keep their
money in savings, and others will borrow money from him. The charts below describe
the amounts that students are saving and borrowing from him. What is the least
possible value, in dollars, of the money that Rohan is currently keeping in his bank?
Express your answer to the nearest whole number.
Amount put in for savings
Value of accounts
Number of accounts
$0.01-$20.00
3
$20.01-$40.00
4
$40.01-$60.00
6
$60.01-$80.00
2
Amount taken for loans
Value of loan
Number of loans
$0.01-$20.00
5
$20.01-$40.00
3
$40.01-$60.00
1
$60.01-$80.00
1
3. ________ A group of N students, where N < 50, is on a field trip. If their teacher puts them
in groups of 8, the last group has 5 students. If their teacher instead puts them in
groups of 6, the last group has 3 students. What is the sum of all possible values of
N?
4. ________ An environmental agency needs to hire a number of new employees so that 85 of the
new employees will be able to monitor water pollution, 73 of the new employees will
be able to monitor air pollution and 27 of the new employees will be able to monitor
both. What is the minimum number of employees that need to be hired?
5. ________ When the length of an object is given to the nearest quarter-inch, the measurement
could be off by a maximum of what part of an inch? Express your answer as a
common fraction.
6. ________ What is the sum of all integer values of x such that [ − is an integer?
7. ________ A palindrome is a number that reads the same forward as backward. How many
3-digit palindromes are multiples of 3?
8. ________ The 52-digit whole number N = 200…003 consists of fifty zeros between the digits
2 and 3. How many digits are zero in the decimal expansion of 1 ?
9. ________ The product of the base seven numbers 247 and 307 is expressed in base seven.
What is the base seven sum of the digits of this product?
10. _______ Let a, b, c, d and e be whole numbers. The sum of the four
numbers on each of the five longest line segments of the
star is the same. If a = 18, what is the value of b?
MATHCOUNTS 2002–03
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Warm-Up & Solutions
1.
Area rec tan gle = 12 • 6 = 72
h=24
base
Area triangle = 72 = 1 base • height = 1 base • 24
2
2
Therefore: base = 144 = 6
6
24
2. Key is to understand: “the least possible value”. This
means having the lowest savings, (deposits) and the
highest loans. A table helps to organize the data:
Savings (deposits or input)
Loans (output)
Range Number Minimum Range Number Maximum
Amount
Amount
0.013
$0.03
0.015
$100.00
20.00
20.00
20.014
$80.04
20.013
$120.00
40.00
40.00
40.016
$240.06 40.011
$60.00
60.00
60.00
60.012
$120.02 60.011
$80.00
80.00
80.00
Total
$440.15
Total
$360.00
Difference: $440.15 - $360.00=$80.15 or $80
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3. Facts: Number of students, N, < 50. Grouped by 8’s
there are 5 left over, grouped by 6’s there are 3 leftover.
N + 5 and N + 3 are the groups
8
6
Make two lists:
N for Groups of 8
N for Groups of 6
45
45
37
39
29
33
21
27
13
21
5
15
9
3
45 and 21 are the only two common solutions. They
asked for the sum of all possible: 45 + 21 = 66
4. This find of problem is best solved using a Venn
diagram: WP=85, AP=73, and there are 27 for both:
WP
AP
27
58
46
WP only = 85 − 27 = 58
AP only = 73 − 27 = 46
Minimum = 58 + 27 + 46 = 131
131
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5. Length is given to nearest ¼ of an inch. What is the
maximum error or the amount that it could be off?
1/4
1/2
3/4
INCH
To either side, the next increment would be used. Right
in the middle, the error is ½ of the increment or 1/8 of an
inch.
1/8
6. The sum of all integer values of X that satisfy:
67
Key is that 67 is a prime so its factors are 67
2x − 23
and 1. Because it is not limited to positive integers, the
results can be negative as well, +/-67 ans +/-1:
2x − 23 = 67 2x = 90 x = 45
2x − 23 = 1 2x = 24 x = 12
2x − 23 = −67 2x = −44 x = −22
2x − 23 = −1 2x = 22 x = 11
45 + 12 + (−22) + 11= 46
46
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7. Palindrome: Number that reads the same forward and
backward, example is 101. Need to find all 3 digit
palindromes that are multiples of 3. To be a multiple of
three, the sum of the digits must add to a multiple of
three:
Strategy, list all palindromes and then select those that
are a multiple of 3:
101, 111, 121, 131, 141, 151, 161, 171, 181, 191: 3 for
100’s
Continue this for 200 through 900 and we get 30
numbers: 30
8. A 52-digit whole number, N, has 50 0’s between the 2
and 3: 200…003. How many zeros in N2 ?
Turn it into a simpler case with just two zeros: 2003
times 2003 and we find the answer 4012009. Notice that
the two zeros repeats: “2009” and there is another 0 in
“401”. What happens if the numbers are 20003:
400120009. The zeros repeat and there is another term
with 1 less zero. Therefore for 50 zeros there will be 50 +
49 or 99 zeros:
99
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