**A. 456.** The point *D* lies in the triangle *ABC* such that the circles inscribed in triangles *ABD*, *BCD*, and *CAD* pairwise touch each other. On lines *BC*, *CA*, *AB*, *AD*, *BD*, *CD*, denote the touching points by *A*_{1}, *B*_{1}, *C*_{1}, *A*_{2}, *B*_{2}, *C*_{2}, respectively. Let lines *B*_{1}*C*_{2} and *B*_{2}*C*_{1} meet at *E*, and let lines *A*_{1}*C*_{2} and *A*_{2}*C*_{1} meet at *F*. Show that the lines *AF*, *BE*, and *C*_{1}*D* are concurrent.

(5 points)

**C. 945.** On 1 March, 2008, the Hungarian National Bank withdrew 1 and 2-forint (HUF) coins from circulation. At the cash desk, prices of individual items are not rounded, but the total is rounded to the nearest five. An information leaflet formulates the rule of rounding as follows: Totals ending in 1 or 2 are rounded down to 0; totals ending in 3 or 4 are rounded up to 5; totals ending in 6 or 7 are rounded down to 5; totals ending in 8 or 9 are rounded up to 0. Martin buys two croissants in the little shop on the corner every morning. In a few days, he saves exactly the price of a croissant. Given that a croissant costs more than 10 forints, what may be the price of a croissant so that he can save that amount in the shortest possible time?

(5 points)