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lfdahl
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Find the minimal value of the expression $\frac{a}{b}$ over all triples $(a, b, c)$ of positive
integers satisfying $|a^c − b!| ≤ b$.
integers satisfying $|a^c − b!| ≤ b$.
lfdahl said:Find the minimal value of the expression $\frac{a}{b}$ over all triples $(a, b, c)$ of positive
integers satisfying $|a^c − b!| ≤ b$.
Albert said:I guess (by instinct) the answer should be $\dfrac {1}{2}$
Am I right ?
I am thinking a method to prove it
my way of thinking:lfdahl said:You are right indeed!
The expression |ac−b|≤b
means to find the smallest possible value of |ac−b|
that is less than or equal to b
.
To solve for the minimal value in the expression |ac−b|≤b
, you can use algebraic methods to manipulate the equation and isolate the absolute value term. Then, you can set up inequalities to find the range of values that ac−b
can take, and determine the smallest value within that range that satisfies the inequality.
For example, if we have the expression |2x−3|≤5
, we can isolate the absolute value by dividing both sides by 2 to get |x−3/2|≤5/2
. Then, we can set up inequalities to find the range of values for x
and determine the smallest value within that range that satisfies the inequality.
Finding the minimal value in the expression |ac−b|≤b
can be useful in various mathematical applications, such as optimization problems or finding the roots of a polynomial function. It helps to identify the smallest value that satisfies a certain condition, which can be helpful in making decisions or solving problems.
Yes, there are a few special cases to consider when solving for the minimal value in the expression |ac−b|≤b
. One case is when b
is equal to 0, in which the minimal value would also be 0. Another case is when c
is equal to 0, in which the minimal value would depend on the value of a
. Careful consideration and analysis of these cases is necessary to arrive at the correct solution.