Problem 20 · 2024 Math Kangaroo
Hard
Algebra & Patterns
sum-constraintcasework
Vlado took part in 31 cross-country races over the last five years. In the first year he ran the fewest races, and the number then increased every year. In the fifth year he ran three times as many races as in the first year. How many races did he run in the fourth year?
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Answer: C — 8
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Hint 1 of 2
Let the first year be a; then the fifth year is 3a, and all five years strictly increase.
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Hint 2 of 2
The five increasing numbers add to 31; test small values of a.
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Approach: bound the first year and pin down the fourth
- With first year a and fifth year 3a, the five strictly increasing counts sum to 31.
- Testing values forces a = 3 (so fifth year 9), with the middle three summing to 19.
- Every valid arrangement puts 8 races in the fourth year.
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