## Subject 4.

Consider the sets $A=\{a, b, c, d\} \subset \mathbb{N}$ and $S=\{s \in \mathbb{N} \mid s=x+y, x \in A, y \in A, x \neq y\}$.

It is known that $S=\{82,96,104,112,126\}$.

a) Calculate the sum of the elements in set $A$;

b) Determine the elements of set $A$.

Prof. Mircea Fianu, Bucharest

| Details of solution | Associated grading |
| :--- | :---: |
| We can consider $a$ as the smallest number in $S$, so $c+d=126$. | $\mathbf{1 p}$ |
| The sum of the elements in $A$ is equal to $(a+b)+(c+d)=82+126=208$. | $\mathbf{1 p}$ |
| b) Since adding different elements from $A$ two at a time results in 6 sums, and set $S$ has 5 elements, it means that two of the 6 sums are equal. | $\mathbf{1 p}$ |
| Since $a+b$ and $b+d=112$. | $\mathbf{1 p}$ |
| Finally, $a=37, b=45, c=59, d=67$. | $\mathbf{2 p}$ |

## MATHEMATICS OLYMPIAD

 - SECTOR STAGE, 09.02.2013 -
## GRADE 5

## Note: All subjects are mandatory. Each subject is graded from 0 to 7 points. Complete solutions must be written on the exam sheet. Time allowed: 2 hours.
4. Consider the sets $A=\{a, b, c, d\} \subset \mathbb{N}$ and $S=\{s \in \mathbb{N} \mid s=x+y, x \in A, y \in A, x \neq y\}$. It is known that $S=\{82,96,104,112,126\}$.

a) Calculate the sum of the elements in the set $A$;

b) Determine the elements of the set $A$.