Difference between revisions of "2018 AMC 10A Problems/Problem 18"
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Note that all numbers formed from this sum are either positive, negative or zero. The number of positive numbers formed by this sum is equal to the number of negative numbers formed by this sum, because of symmetry. There is only one way to achieve a sum of zero, if all <math>a_i=0</math>. The total number of ways to pick <math>a_i</math> from <math>i=1, 2, 3, ... 7</math> is <math>3^8=6561</math>. <math>\frac{6561-1}{2}=3280</math> gives the number of possible negative integers. The question asks for the number of nonnegative integers, so subtracting from the total gives <math>6561-3280=\boxed{3281}</math>. | Note that all numbers formed from this sum are either positive, negative or zero. The number of positive numbers formed by this sum is equal to the number of negative numbers formed by this sum, because of symmetry. There is only one way to achieve a sum of zero, if all <math>a_i=0</math>. The total number of ways to pick <math>a_i</math> from <math>i=1, 2, 3, ... 7</math> is <math>3^8=6561</math>. <math>\frac{6561-1}{2}=3280</math> gives the number of possible negative integers. The question asks for the number of nonnegative integers, so subtracting from the total gives <math>6561-3280=\boxed{3281}</math>. | ||
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+ | ~RegularHexagon |
Revision as of 18:25, 8 February 2018
How many nonnegative integers can be written in the form
where
for
?
Solution
This looks like balanced ternary, in which all the integers with absolute values less than are represented in
digits. There are 8 digits. Plugging in 8 into the formula gives a maximum bound of
, which means there are 3280 positive integers, 0, and 3280 negative integers. Since we want all nonnegative integers, there are
integers or
.
Solution 2
Note that all numbers formed from this sum are either positive, negative or zero. The number of positive numbers formed by this sum is equal to the number of negative numbers formed by this sum, because of symmetry. There is only one way to achieve a sum of zero, if all . The total number of ways to pick
from
is
.
gives the number of possible negative integers. The question asks for the number of nonnegative integers, so subtracting from the total gives
.
~RegularHexagon