Problem 14 · 2014 Math Kangaroo
Medium
Number Theory
casework
How many whole-number triples \((a,b,c)\) with \(a>b>c>1\) fulfil the condition \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1\)?
Show answer
Answer: C — 2
Show hints
Hint 1 of 2
With a > b > c > 1 the smallest the values can be is c = 2, b = 3, a = 4.
Still stuck? Show hint 2 →
Hint 2 of 2
Even those nearly-largest fractions barely beat 1, so very few triples can work.
Show solution
Approach: push the values to their smallest and count
- Since c > 1 and the values are distinct decreasing, the only candidates start at c = 2, b = 3.
- (4,3,2): 1/4+1/3+1/2 = 13/12 > 1 ✓; (5,3,2): 1/5+1/3+1/2 = 31/30 > 1 ✓.
- Any larger a (with b=3,c=2) or any larger b drops the sum to 1 or below.
- So exactly 2 triples work.
Mark:
· log in to save