Subject 2. The arithmetic mean of the distinct natural numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$, 2014, and 2016 is 2015.

a. Find the arithmetic mean of the numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$.

b. If the numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2014}$ are even, show that $a_{1} \cdot a_{2} \cdot a_{3} \cdot \ldots \cdot a_{2014}=0$
## Subject 2.

| a. | $\left(a_{1}+a_{2}+a_{3}+\ldots+a_{2014}+2014+2016\right): 2016=2015 \Rightarrow$  $a_{1}+a_{2}+a_{3}+\ldots+a_{2014}=2015 \cdot 2016-2016-2014 \Rightarrow$  $\left(a_{1}+a_{2}+a_{3}+\ldots+a_{2014}\right): 2014=2015$ | 3p |
| :--- | :--- | :---: |
|  | Calculate the sum of the smallest 2014 distinct, non-zero even numbers, different from 2014 and 2016: $2+4+6+\ldots+2012+2018+\ldots+4032=2(1+2+\ldots+2016)-2014-2016=$  $=2016 \cdot 2017-2016-2014=4062242$ which is 4032 more than 2014$\cdot$2015. We need to subtract 4032 from the sum, the numbers must remain even, so one of the numbers becomes 0. | 4p |