Find the largest possible value of n, such that there exists n real numbers x1,...,xn which satisfy |xi−xj|>1/100(1+xixj) for all values of i≠j.
To find the largest possible value of n, let's consider the given condition:
|xi−xj| > 1/100(1 + xi*xj)
Let's start by simplifying the condition:
100|xi−xj| > 1 + xi*xj
100|xi−xj| - xi*xj > 1
Now, let's consider the different cases for xi and xj:
1. If xi = xj, then the condition simplifies to 0 > 1, which is not possible.
2. If xi > xj, then the condition becomes:
100(xi−xj) - xi*xj > 1
Dividing both sides by xi - xj (since xi > xj), we get:
100 - xj > (1 / (xi - xj))
Since xi > xj, the denominator is positive, so we can multiply both sides by (xi - xj) without changing the inequality:
100(xi - xj) - xi*xj(xi - xj) > 1
Simplifying further:
100(xi - xj) - xi^2*xj + xj^2*xi > 1
Multiplying out the terms:
100xi - 100xj - xi^2*xj + xj^2*xi > 1
Now let's consider the case when xi < xj (the inequality will have a similar form):
100xj - 100xi - xj^2*xi + xi^2*xj > 1
Now, let's combine the two cases and rewrite the inequality:
100xi - 100xj - xi^2*xj + xj^2*xi > 1
100xj - 100xi - xj^2*xi + xi^2*xj > 1
Combining the terms:
100(xi - xj) - xi^2*xj + xj^2*xi > 1
Since this condition must hold for all values of xi and xj, we can consider the worst-case scenario where xi and xj take extreme values. Let's consider when xi = 1 and xj = -1 to maximize the difference between the terms:
100(1 - (-1)) - 1^2*(-1) + (-1)^2*1
= 100(2) - (-1) + 1
= 200 + 1 + 1
= 202
Therefore, the largest possible value of n is 202.
To find the largest possible value of n, we need to determine the maximum number of real numbers x1,...,xn that satisfy the given condition. Let's break down the problem step by step:
1. First, let's rewrite the given condition:
|xi - xj| > 1/100(1 + xi*xj)
2. Next, we notice that if we consider the worst-case scenario where the right side of the inequality is as small as possible, we can find an upper bound. This is because if the condition is satisfied for the smallest possible right-hand side value, it will hold true for any larger value.
Let's assume:
1/100(1 + xi*xj) = ε, where ε is a very small positive number (close to zero).
Rearranging the equation gives us:
|xi - xj| > 100ε
Now, we can formulate a plan to find an upper bound:
- We will select the largest possible value for xi (let's call it x_max).
- We will assume that all other values, x1 to xn-1, are equal to x_max. This maximizes the difference |xi - xj|.
3. With this plan in mind, let's substitute xi = x_max and xj = x_max into the inequality:
|x_max - x_max| > 100ε
0 > 100ε
Since ε is a very small positive number, let's assume it is expressed as ε = 1/10100.
0 > 100(1/10100)
0 > 1
The inequality 0 > 1 is false. Therefore, there is no possible value of x_max that satisfies the condition.
4. Based on our analysis, we conclude that there are no real numbers x1, ..., xn that satisfy the given condition. Hence, the largest possible value of n is 0.
Therefore, the answer to the question is n = 0.