## SUBJECT 1

a) Show that the equality $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$ is satisfied for $x=14$ and $y=58$.

Give another example of natural numbers for which the equality is satisfied.

b) If $x$ and $y$ are positive rational numbers and $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$, calculate the value of the expression $\frac{x}{x+1}+\frac{y}{y+2}$.
SUBIECTUL 1

a) Show that the equality $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$ is verified for $x=14$ and $y=58$. Provide another example of natural numbers for which the equality is verified.

| a) Verification | $1 p$ |
| :--- | :---: |
| An example | $2 p$ |
| Examples:  (10, 218), (11, 118), (13, 68), (14, 58), (17, 43), (19, 38), (29, 28), (34, 26), (49, 23), (59, 22), (109, 20),  $(209 ; 19)$. |  |
| b) $\frac{x}{x+1}+\frac{y}{y+2}=\frac{x+1-1}{x+1}+\frac{y+2-2}{y+2}$ | $2 p$ |
| $=2-\left(\frac{1}{x+1}+\frac{2}{y+2}\right)$ | $1 p$ |
| $=2-\frac{1}{10}=\frac{19}{10}$. | $1 p$ |

OBS: If the student chooses to add to both sides of the equality $\frac{1}{x+1}+\frac{2}{y+2}=\frac{1}{10}$ the expression $\frac{x}{x+1}+\frac{y}{y+2}(2 p)$.

Then if they write $2=\frac{1}{10}+\frac{x}{x+1}+\frac{y}{y+2}(1 p)$ and $\frac{x}{x+1}+\frac{y}{y+2}=\frac{19}{10}(1 p)$.