Arithmetic string equal to the geometric string, count x

[ghostfirefox]

Let x1, ..., x25 be such positive integers that x1⋅x2⋅ ... ⋅x25 = x1 + x2 + ... + x25. What is the maximum possible value of the largest of numbers x1, x2, ..., x25?

 

[jvicens]

The maximum possible value of the largest number in this scenario is 25. This can be achieved by setting all 25 numbers to 1, as 1+1+...+1 = 25 and 1*1*...*1 = 25.

To prove that this is the maximum possible value, let's assume that there exists a larger number, say n, in the set of x1, x2, ..., x25. This would mean that n is greater than 1, since we have already established that 1 is the maximum value.

Since n is a positive integer, it can be written as a sum of positive integers, let's say n = a + b, where a and b are positive integers.

Now, let's consider the product of all 25 numbers: x1 * x2 * ... * x25. From the given condition, we know that this product is equal to the sum of all 25 numbers: x1 + x2 + ... + x25.

Since n is one of the numbers in this product and sum, we can replace it with a + b. This gives us:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25

Replacing n with a + b, we get:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25 - n + n

Since n = a + b, we can rewrite this as:

x1 * x2 * ... * x25 = x1 + x2 + ... + x25 - (a + b) + (a + b)

Now, we can rearrange the terms in the sum on the right-hand side to get:

x1 * x2 * ... * x25 = (x1 - a) + (x2 - b) + ... + (x25 - b) + (a + b)

We know that n = a + b and n is one of the numbers in the sum on the right-hand side. Therefore, we can replace n with a + b to get:

x1 * x2 * ... * x25 = (x1 - a) + (x2 - b) + ... + (x25 - b) + n

Since n is greater than 1, we know that at least one of the numbers in the sum on