Mathcounts Chapter Competition Solutions Worksheet - Middle School - 2014 Page 3
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12triangle with side length s, has hypotenuse of
If you are familiar with the concept of
length s√2. Thus, the length of the diagonal
combinations, you can solve this problem by
finding the number of combinations of 2
is 20√2 Ans.
objects chosen from a collection of 6 objects.
If you aren’t familiar with this property of
That’s
C
= 6!/(4! × 2!) = 30/2 = 15 Ans.
454590 right triangles, you can use the
6
2
Pythagorean Theorem. Let x represent the
length of the diagonal.
12. According to the bar graph
20
2
+ 20
2
= x
2
2 workers earn $41,000 to $50,000.
400 + 400 = x
2
6 workers earn $51,000 to $60,000.
x
2
= 800
5 workers earn $61,000 to $70,000.
x = √800 = √2 × √4 × √100
4 workers earn $71,000 to $80,000.
x = √2 × 2 × 10 = 20√2 Ans.
2 workers earn $81,000 to $90,000.
To find the greatest mean salary we need to
take the highest value in each range, multiply
9. a ۞ b = ab/(a + b)
it by the number of workers in that range, add
4a/(a + 4) = 3
them all up and divide by the number of
4a = 3(a + 4)
workers. Then we need to follow the same
4a = 3a + 12
steps to find the lowest mean salary. Let’s see
a = 12 Ans.
if we can avoid actually adding and multiplying
those large values since we do have the aid of
10. A jar has 3 green, 2 red and 5 blue marbles.
a calculator. Let’s write the sum of the lowest
The jar has a total of
salaries as
3 + 2 + 5 = 10 marbles.
2a + 6b + 5c + 4d + 2e = S. Since the
P(red) = 2/10
difference between the lowest and highest
P(blue) = 5/10
salary in each range is $9000, the sum of the
P(red or blue) = 2/10 + 5/10
highest salaries is 2(a + 9000) + 6(b +
= 7/10 Ans.
9000) + 5(c + 9000) + 4(d + 9000) + 2(e +
9000)) = 2a + 6b + 5c + 4d + 2e + 9000(2
+ 6 + 5 + 4 + 2) = S + 9000(19) = S +
11. Six points are in a plane and no three are
171,000. The difference in the means would
collinear. Find the number of distinct lines
be ((S + 171,000) − S)/19 = 171,000/19 =
that are determined by all possible pairs of
9000 Ans.
points.
There are 6 points and each point connects to
5 other points making a total of 6 × 5 = 30
13. The sum of the first n integers is (n + 1) ×
lines. But we end up counting the same line
(n/2). So, the sum of the first 2000 positive
twice, as this example illustrates:
integers is (2000 + 1) × (2000/2) = 2001 +
1
1
1000 = 2,001,000 Ans.
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14. A stalactite grows downward at 0.004 inches
The line from point 1 to point 4 is counted
per year. A stalagmite, directly below grows
once among the lines between point 1 and
upward at 0.006 inches per year. The height
the 5 other points, then again with the lines
of the ceiling is 10 feet (or 120 inches). Each
from point 4 to the other 5 points. This will be
passing year, the stalactite and stalagmite
true for each line. So, 30/2 = 15 Ans.
grow 0.006 + 0.004 = 0.01 inches closer. At
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