4. Consider the numbers $A=1+2+3+\ldots+2016^{2}$ and $B=2+4+6+\ldots+2016^{2}$.

a) Find the last digit for each of the numbers $A$ and $B$.

b) Show that $A-B$ is a perfect square.

The problems were proposed by: Prof. Mariana Guzu, School "D. Zamfirescu" Prof. Marius Mohonea, C.N. "Unirea"

NOTE: Working time: 2 hours.

Each problem is graded from 0 to 7 points.

## NATIONAL MATHEMATICS OLYMPIAD

Local stage - 20.02.2016

5th Grade

Grading and marking guide
4. a) $A=1008 \cdot 2016 \cdot\left(2016^{2}+1\right)$ ..... $1 \text{p}$
$A=\overline{\ldots 8} \cdot \overline{\ldots 6} \cdot \bar{\ldots} \bar{\ldots}=\overline{\ldots 6}$ ..... $1 \text{p}$
$B=1008 \cdot 2016 \cdot(1008 \cdot 2016+1)$ ..... $1 \text{p}$
$B=\overline{\ldots 8} \cdot \overline{\ldots 6} \cdot \overline{\ldots 9}=\overline{\ldots 2}$ ..... $1 \text{p}$
b) $\quad A-B=(1008 \cdot 2016)^{2}$ ..... $3 \text{p}$

NOTE. For any correct solution, even if it is different from the one in the grading rubric, the corresponding score is awarded.