Mathcounts Chapter Competition Solutions Worksheet - Middle School - 2014 Page 7
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12Subtracting the second equation from the first
than each of the other two boxes?
yields the following:
If we transfer 6 marbles from B to C, then B
(x
2
+ z
2
) − (w
2
+ x
2
) = 49 − 25
will have 146 marbles and A and C will each
z
2
− w
2
= 24
have 142 marbles.
If we transfer 2 more marbles from B to C
To this we can add the third equation to get:
(z
2
− w
2
) + (w
2
+ y
2
) = 24 + 64
then C will have 144 marbles as will B.
y
2
+ z
2
If we transfer one more marble from B to C
= 88
So, DE = √88 = 2√22 Ans.
we will have:
Box A with 142 marbles
Box B with 143 marbles.
30. What is the value of
Box C with 145 marbles.
� 1 + � 2 + � 2 + √ 2 +… where all
The total number of marbles transferred
6 + 2 + 1 = 9 Ans.
subsequent numbers in the expression are
2s?
2. What is the largest prime that divides both
Start by finding the value of the square of the
20! + 14! and 20! – 14!?
expression. The square is
The expression 20! + 14! can be rewritten as
1 + � 2 + � 2 + � 2 + √2 +…
20 × 19 × 18 × 17 × 16 × 15 × 14! + 14!
= 14!((20 × 19 × 18 × 17 × 16 × 15) + 1).
Similarly, 20! – 14! can be rewritten as
Let x = � 2 + � 2 + � 2 + √ 2 +…
20 × 19 × 18 × 17 × 16 × 15 × 14! − 14!
= 14!((20 × 19 × 18 × 17 × 16 × 15) − 1).
√ 2 + x = x
The greatest common factor for 20! + 14!
and 20! – 14! is 14!, and 13 is the largest
prime factor of 14! So the greatest common,
2 + x = x
2
prime factor of these two numbers is 13 Ans.
x
2
− x − 2 = 0
(x + 1)(x − 2) = 0
x = −1 or x = 2
3. In how many ways can the letters in the word
Since x must be positive, x = 2.
NINE be arranged?
1 + � 2 + � 2 + � 2 + √2 +… = 1 + 2 = 3
If all four letters were different the answer
would be 4! or 24. But since two of the letters
are the same, we have to account for
duplicates. So the total number of
Recall that the original expression is the
arrangements is 24/2 = 12 Ans.
square root of this expression. So, the value of
the original expression is √3 Ans.
4. A vendor offers discounts for orders of 11 or
more shirts as follows:
Target Round
11-25 – 10% off
1. Box A has 142 marbles.
26-50 – 15% off
Box B has 152 marbles.
51-100 – 20% off
Box C has 136 marbles.
101-250 – 30% off
Moving marbles only from B to C, what’s the
251 or more – 35% off
least number of marbles that need to be
For how many different quantities of shirts
transferred so that C contains more marbles
would the cost exceed the cost of buying the
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