Problem 11 · 2016 Math Kangaroo
Stretch
Algebra & Patterns
substitution
For the real numbers a, b, c, d the following holds true: a + 5 = b2 − 1 = c2 + 3 = d − 4. Which of the numbers a, b, c, d is biggest?
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Answer: D — d
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Hint 1 of 2
Set the common value to k and write each letter in terms of k.
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Hint 2 of 2
Compare a = k−5, d = k+4, and the square-root sizes of b and c.
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Approach: express through one value
- Let the equal value be k. Then a = k−5, d = k+4, b2 = k+1, c2 = k−3.
- d exceeds a by 9, and d = k+4 also beats |b| = √(k+1) and |c| = √(k−3) for every valid k.
- So d is the largest.
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