## Subject IV.

Determine the natural numbers of the form $\overline{a b c}$ that satisfy the relation $\overline{a b} \cdot\left(b^{2}+c^{2}\right)=2014$

Mathematical Gazette

## Note.

All problems are mandatory.

Each problem is worth 7 points.

Time allowed: 2 hours.

Proposers: Todoran Nicoleta, Seceleanu Daniela, Ginta Florica.

## MURES COUNTY SCHOOL INSPECTORATE

## S.S.M.R - MURES BRANCH  Mathematics Olympiad  Local phase 14.02.2014  5th Grade  Grading Rubric
## SUBJECT IV

Determine the natural numbers of the form $\overline{a b c}$ that satisfy the relation $\overline{a b} \cdot\left(b^{2}+c^{2}\right)=2014$

Mathematical Gazette

## Solution

$2014=2 \cdot 19 \cdot 53$ (1p) $\Rightarrow \overline{a b} \cdot\left(b^{2}+c^{2}\right)=2 \cdot 19 \cdot 53$ (1p)

## Case I

$$
\overline{a b}=19 \Rightarrow c^{2}=106-81 \Rightarrow c=5 \Rightarrow \overline{a b c}=195 \text { (3p) }
$$

## Case II

$$
\overline{a b}=38 \Rightarrow c^{2}=53-64 \Rightarrow c \notin N
$$

## Case III

$$
\overline{a b}=53 \Rightarrow c^{2}=38-9 \Rightarrow c=29 \text { is not a digit (1p) }
$$