Subject 4. The angles $\varangle A O B, \varangle B O C, \varangle C O A$ are angles around a point such that $\varangle A O B \equiv \varangle A O C$ and the complement of angle $\varangle B O C$ is equal to the supplement of angle $\varangle A O B$.

a) Determine the measures of angles $\varangle A O B$ and $\varangle B O C$.

b) Let $[O D$ be the semiline opposite to semiline $[O A$ and points $M \in(O B$, $N \in(O C$ and $P \in(O D$ such that $[O M] \equiv[O N]$. Prove that $[M P] \equiv[N P]$.

## Note: All subjects are mandatory

Time for work: 2 hours

## MATHEMATICS OLYMPIAD  LOCAL STAGE  JANUARY 30, 2016  GRADE 6  Scoring Guide
## Subject 4.

| a. | Let $m(\Varangle A O B)=x$ and $m(\Varangle B O C)=y \Rightarrow 2 x+y=360^{\circ}$. | $\mathbf{1 p}$ |
| :--- | :--- | :--- |
|  | The second condition: $90^{\circ}-y=180^{\circ}-x \Leftrightarrow x=y+90^{\circ}$. | $\mathbf{1 p}$ |
|  | Substituting into the first relation, we have $2\left(y+90^{\circ}\right)+y=360^{\circ} \Leftrightarrow 3 y=180^{\circ} \Leftrightarrow$ |  |
|  | $m(\Varangle B O C)=60^{\circ} \Rightarrow m(\Varangle A O B)=150^{\circ}$ | $\mathbf{1 p}$ |
| b. | $\varangle M O P \equiv \varangle N O P-$ supplements of congruent angles | $\mathbf{1 p}$ |
|  | Congruence of triangles $\triangle M O P \equiv \triangle N O P$ (SAS) | $\mathbf{2 p}$ |
|  | Conclusion $[M P] \equiv[N P]$ | $\mathbf{1 p}$ |