[ Arithmetic. Mental calculation, etc.]

What is the value of the expression $\left(10^{2}+11^{2}+12^{2}+13^{2}+14^{2}\right) / 365$ ?
Notice that one can simply "calculate directly," i.e., compute each of the squares, add them all up, and finally divide; or one can recall the formula for the square of a sum and make some transformations.

## Solution

Let's outline the steps that can quite easily be done mentally.

$$
10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=10^{2}+(10+1)^{2}+(10+2)^{2}+(10+3)^{2}+(10+4)^{2}=5 \times 10^{2}+2 \times 10 \times(1+2+3+4)+1^{2}+2^{2}
$$

$+3^{2}+4^{2}=500+200+1+4+9+16=730$.

Now it's easy to figure out that the answer to the problem is 2. The problem can also be solved differently:

$10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=(12-2)^{2}+(12-1)^{2}+12^{2}+(12+1)^{2}+(12+2)^{2}=5 \times 12^{2}+2 \times 10 \times(1+2-1-2)+2 \times\left(1^{2}+2^{2}\right)=$ $60 \times 12+0+2 \times 5=720+10=730$.

And now it's also easy to figure out that the answer to the problem is 2.