1985 USAMO Problems/Problem 2
Determine each real root of
correct to four decimal places.
The equation can be re-written as
We first prove that the equation has no negative roots. Let The equation above can be further re-arranged as The right hand side of the equation is negative. Therefore and we have Then the left hand side of the equation is bounded by However, since and it follows that for negative Then The right hand side of the equation is then a large negative number. It cannot be equal to the left hand side which is bounded by 9.
Now let When the left hand side of equation (1) is negative. Therefore the equation has real roots on both side of , as its leading coefficient is positive. We will prove that is a good approximation of the roots (within ). In fact, we can solve the "quadratic" equation (1) for : Then Easy to see that for positve Therefore, Then
Let be a root of the equation with Then and An aproximation of is defined as follows: We check the error of the estimate:
The first absolute value
The second absolute value through a rationalized numerator.Therefore
For a real root with we choose We can similarly prove it has the desired approximation.
Another round of iteration can increase the accuracy to more than 10 decimal places:
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