AustriaMO2013

We first note that $45^2=2025$. For any number $n$, the number of divisors less than $\sqrt{n}$ is certainly equal to the number of divisors greater than $\sqrt{n}$, since $0<t<\sqrt{n}$ implies $\frac{n}{t}>\sqrt{n}$ and $t|n$ implies $\frac{n}{t}|n$ (and vice versa).

For all numbers from $2000$ through $2100$, we have $44<\sqrt{n}<46$. For all of these numbers, the number of divisors less than $45$ is therefore equal to the number of divisors greater than $45$. It follows that the probability of a random number being not greater than $45$ is equal to $\frac{1}{2}$ for all $n\ne 2025$.

For $n=2025$, $45$ is also a divisor, but since $\frac{n}{t}=t$ in this case, the probability for $2025$ is greater than $\frac{1}{2}$, and $2025$ is therefore the number with the required property. $\square$