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, we need to consider the conditions given in the problem and determine the maximum number of real numbers that satisfy those conditions.
First, let's simplify the given inequality:
|xi − xj| > 1/100(1 + xi · xj)
Since we are interested in maximizing n, let's assume that x1, x2, x3, ... xn are all distinct real numbers. This assumption will help us to find the largest possible value of n.
Now, let's consider two cases:
Case 1: xi ≠ -1 (for all i)
In this case, we can multiply both sides of the inequality by (1 + xi · xj) to obtain:
(1 + xi · xj) · |xi − xj| > 1/100
Let's rewrite the left side of the inequality in terms of differences of squares:
(xi2 − xj2) + xi · xj (xi − xj) > 1/100
Now, let's separate the terms:
(xi + xj) · (xi − xj) + xi · xj (xi − xj) > 1/100
Factoring out the common term:
(xi − xj) · (xi + xj + xi · xj) > 1/100
Since xi ≠ xj (as assumed), we can divide both sides by (xi − xj):
xi + xj + xi · xj > 1/100
To find the maximum possible value of n, let's find the minimum value of xi + xj + xi · xj.
Considering the terms xi, xj, and xi · xj, we want to maximize all three individually. For xi and xj, the maximum values occur when they are both equal to the largest possible real number, which is positive infinity (∞).
Now let's consider xi · xj. Since we want to maximize it as well, we need to maximize both xi and xj. This means both xi and xj need to be positive infinity (∞).
Substituting these values into xi + xj + xi · xj > 1/100:
∞ + ∞ + ∞ · ∞ > 1/100
This inequality simplifies to:
2∞ > 1/100
But we know that 2∞ = ∞, so we can rewrite the inequality as:
∞ > 1/100
This is always true. Consequently, there is no constraint on n in Case 1, meaning n can be any positive integer.
Case 2: xi = -1 (for some i)
If xi = -1, we need to modify the inequality as follows:
|xi − xj| > 1/100(1 − xi · xj)
Using similar logic as in Case 1, we can determine that in this case as well, n can be any positive integer.
In conclusion, the largest possible value of n is infinity (∞) for both cases. This means there is no constraint on the number of real numbers xi that satisfy the given conditions. Therefore, any positive integer value of n can be chosen.