2020 AMC 8 Problems/Problem 16
Each of the points and
in the figure below represents a different digit from
to
Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums is
What is the digit represented by
Solution 1
We can form the following expressions for the sum along each line:
Adding these together, we must have
, i.e.
. Since
are unique integers between
and
, we obtain
(where the order doesn't matter as addition is commutative), so our equation simplifies to
. This means
.
~RJ5303707
Solution 2
Following the first few steps of Solution 1, we have . Because an even number (
) subtracted from an odd number (47) is always odd, we know that
is odd, showing that
is odd. Now we know that
is either 1, 3, or 5. If we try
, we get
which is not true. Testing
, we get
, which is also not true. Therefore, we have
.
Video Solution
https://youtu.be/SPNobOd4t1c (Channel also has resources to prepare for your AIME qualification)
See also
2020 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
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