2015 AIME I Problems/Problem 1
Problem
The expressions = and = are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers and .
Solution 1
We see that
and
.
Therefore,
Solution 2 (slower solution)
For those that aren't shrewd enough to recognize the above, we may use Newton's Little Formula to semi-bash the equations.
We write down the pairs of numbers after multiplication and solve each layer:
and
Then we use Newton's Little Formula for the sum of n terms in a sequence.
Notice that there are 19 terms in each sequence, plus the tails of 39 and 1 on the first and second equations, respectively.
So: (this is as far as my latex knowledge goes)
((19 choose 1) x 2) + ((19 choose 2) x 10) + ((19 choose 3) x 8) + 1
((19 choose 1) x 6) + ((19 choose 2) x 14) + ((19 choose 3) x 8) + 39
Subtracting A from B gives:
((19 choose 1) x 4) + ((19 choose 2) x 4) - 38
Which unsurprisingly gives us
-jackshi2006
See also
2015 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
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