5.1. A random number sensor outputs a number $a$ - one of the natural numbers $1,2, \ldots, 100$ (with equal probability). For this value of $a$, we find the maximum possible value $M$ of the function

$$
f(x)=\frac{700}{x^{2}-2 x+2 a}
$$

The probability that $M>10$ is $n$ percent. What is $n$?
Answer: 35

Solution. The minimum value of the quadratic trinomial $x^{2}-2 x+2 a$ is $2 a-1$. Since for the considered values of $a$ we have $2 a-1>0$ (which is true for $a>1 / 2$), the maximum value of $f(x)$ is $M=\frac{700}{2 a-1}$. Next, the condition $M>10$ (for $2 a-1>0$) is equivalent to the following conditions: $700>10(2 a-1) \Leftrightarrow$ $71>2 a$. Among the natural numbers $a=1,2, \ldots, 100$, the last inequality is satisfied by the first 35 values.