Suppose f(x) is a polynomial with integer coefficients of degree 100. Find the biggest possible number of pairs of integers n<m, such that f(n)=m and f(m)=n.

Details and assumptions
You are asked to find the biggest possible number of pairs, not the biggest pair. Hence, your answer is just an integer, not a pair of integers.

To find the biggest possible number of pairs of integers (n, m) such that f(n) = m and f(m) = n, we need to analyze the properties of the polynomial f(x).

Since f(x) is a polynomial of degree 100 with integer coefficients, we know that it can be expressed as:

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀, where aₙ ≠ 0 and n = 100.

Let's start by considering the equation f(n) = m, where n and m are integers. Substituting n into the polynomial, we have:

f(n) = aₙnⁿ + aₙ₋₁nⁿ⁻¹ + ... + a₂n² + a₁n + a₀ = m.

Similarly, if we substitute m into the polynomial, we get:

f(m) = aₙmⁿ + aₙ₋₁mⁿ⁻¹ + ... + a₂m² + a₁m + a₀ = n.

To understand the behavior of the polynomial, let's consider two cases:

Case 1: If f(x) is an increasing function
In this case, as x increases, f(x) also increases. Therefore, if n < m, we can conclude that f(n) < f(m). Since f(n) = m and f(m) = n, this case is not possible.

Case 2: If f(x) is a decreasing function
In this case, as x increases, f(x) decreases. Therefore, if n < m, we have f(n) > f(m). If we assume that f(n) = m, then f(m) = f(f(n)) < f(n) = m, which leads to a contradiction.

From the analysis above, we can conclude that there are no pairs of integers (n, m) such that f(n) = m and f(m) = n, when f(x) is a polynomial of degree 100 with integer coefficients. Thus, the answer is 0.