Problem 14 · 2019 Math Kangaroo
Hard
Fractions, Decimals & Percents
proportion
The numbers a, b, c and d are pairwise different integers between 1 and 10 (1 and 10 included). What is the smallest possible value of the expression ab + cd ?
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Answer: C — 1445
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Hint 1 of 2
To make a/b + c/d small, use small numerators and large denominators.
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Hint 2 of 2
Try numerators 1 and 2 with denominators 9 and 10, paired to minimise the sum.
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Approach: minimise by smart pairing
- Use the small numerators 1, 2 and large denominators 9, 10.
- Pairing 1/9 + 2/10 = 1/9 + 1/5 = 14/45.
- No other choice of distinct 1–10 values beats it, so the minimum is 14/45.
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