Problem 10 · 2014 Math Kangaroo
Medium
Number Theory
digit-sumplace-value
In the year 2014 all digits are different and the last digit is bigger than the sum of the other three digits. How many years ago was this last the case?
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Answer: C — 305
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Hint 1 of 2
Read 2014's two conditions, then walk backwards year by year.
Still stuck? Show hint 2 →
Hint 2 of 2
You need all four digits different AND the last digit larger than the sum of the first three.
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Approach: step backward to the previous year meeting both rules
- 2014 works: digits 2,0,1,4 are all different and 4 > 2+0+1.
- Going back, you must find a year with four different digits whose last digit beats the sum of the other three — hard once the leading digits grow.
- The most recent earlier such year is 1709 (digits all different, 9 > 1+7+0 = 8).
- That was 2014 − 1709 = 305 years ago.
Mark:
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