Let
5 + √23
------- = a+b, where a is a positive
2
integer and b is a real number satisfying 0 ≤ b < 1
Evaluate: a
3 + (3 + √23)b
**** Adapted from a problem appearing at 2017 SMO Junior 2017.
√23 is about 4.79
5+√23
------- = about 4.89791576
2
a=4 and b = 0.89791576; the key point being b > 0.5
which is also clear since 23 is much closer to 25 than to 16.
√23 = (2a-5) + 2b = 3 + 2b
2b > 1 and thus (2a-5) = 3 --> a=4 is confirmed.
b = (√23 - 3)/2
Evaluate: a^3 + (3 + √23)b
a^3 = 64
(3 + √23)b = (√23 + 3)(√23 - 3)/2 = 7
The requested expression is 64 + 7 = 71
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Posted by Larry
on 2024-02-24 11:23:08 |
solution
