Find the largest number of real numbers x1,...,xn such that for all i≠j, |xi−xj|>1/100(1+xi xj).

To find the largest number of real numbers x1, x2, ..., xn that satisfy the given condition, let's analyze the equation step by step:

|xi - xj| > (1/100)(1 + xi * xj)

1. First, rearrange the equation to isolate the absolute value:

(xi - xj) > (1/100)(1 + xi * xj)

2. We want to find the largest number of real numbers that satisfy this inequality for all i ≠ j. To do so, let's determine the worst-case scenario for the variables xi and xj.

Take the maximum value possible for xi and the minimum value possible for xj to create the most challenging condition:

- For xi, take the maximum value, which is positive infinity.
- For xj, take the minimum value, which is negative infinity.

Plugging these values into the inequality:

(positive infinity - negative infinity) > (1/100)(1 + positive infinity * negative infinity)

Since infinity multiplied by negative infinity results in negative infinity, we can rewrite the inequality as:

positive infinity > (1/100)(1 - positive infinity)

Now, we have a clearer form of the inequality:

positive infinity > (1/100)(- positive infinity)

This inequality is true because even if we consider the largest possible values for xi and the smallest possible values for xj, the inequality holds.

Therefore, the inequality holds for any real numbers of xi and xj, as long as i is not equal to j.

3. From the analysis above, we can conclude that there is no limit on the largest number of real numbers x1, x2, ..., xn that satisfy the given inequality. The inequality holds for all real numbers.

Hence, we can have an infinite number of real numbers x1, x2, ..., xn satisfying the given condition.