perplexus.info :: Just Math : Positive Integer and Real Number Expression 2

Positive Integer and Real Number Expression 2 (Posted on 2024-02-24)

Let

5 + √23
------- = a+b, where a is a positive 
   2
integer and b is a real number satisfying  0 ≤ b < 1

Evaluate: a³ + (3 + √23)b

**** Adapted from a problem appearing at 2017 SMO Junior 2017.

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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Solution

| Comment 1 of 2

16 < 23 < 25, so I can say (5+sqrt(16))/2 < (5+sqrt(23))/2 < (5+sqrt(25))/2.

Which simplifies to 4.5 < (5+sqrt(23))/2 < 5

Therefore a=4 and then b = (5+sqrt(23))/2 - 4 = (-3+sqrt(23))/2.

Then a^3 + (3+sqrt(23))*b = 4^3 + (3+sqrt(23))*(-3+sqrt(23))/2 = 64 + (-9+23)/2 = 71.

Posted by Brian Smith on 2024-02-24 10:58:08

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