3. We consider two-digit numbers (so the first digit cannot be 0). Such a number is called vain if the sum of the two digits is at least as large as the product of the two digits. The number 36, for example, is not vain, because $3+6$ is smaller than $3 \cdot 6$.

How many two-digit numbers are vain?
A) 17
B) 18
C) 26
D) 27
E) 37
A3. D) 27 Name the two digits $a$ and $b$, where $a$ is the smallest digit of the two (or $a$ is equal to $b$). The number is vain if $a+b \geqslant a \cdot b$. We will now check in several cases when this holds.

If $a \geqslant 2$, then $a \cdot b \geqslant 2 \cdot b=b+b \geqslant a+b$; this is precisely the opposite inequality. We see that equality can only hold if $a$ and $b$ are equal (since $b+b=a+b$ must hold) and moreover, $a=2$ (since $b>0$ and $a \cdot b=2 \cdot b$ must hold). The only vain number we find this way is 22.

In the case $a \leqslant 1$, $a \cdot b \leqslant b \leqslant a+b$ holds. In this case, the number is therefore always vain. We now count the two-digit numbers where the smallest digit is a 0 or a 1. These are exactly the numbers from 10 to 19 and the numbers 20, 21, 30, 31, ..., 90, 91. That's 26 numbers.

In total, there are thus $1+26=27$ vain two-digit numbers.