## Task 1 - 100711

At a sports festival, a race is to be held between young Pioneers and FDJ members according to the following rules:

At the midpoint of one of the shorter sides of a rectangular playing field ($50 \mathrm{~m} \times 70 \mathrm{~m}$), an FDJ member stands, and at the midpoint of the opposite side, a Pioneer stands. Both are to run on a command to the same flag placed on the playing field, taking the shortest route from their starting positions to the flag.

For this purpose, the flag, when the two runners are in their starting positions, should be placed such that it is $50 \mathrm{~m}$ from the FDJ member and $25 \mathrm{~m}$ from the Pioneer.

Give the number of all possible ways to place the flag on the playing field according to the conditions!
The corners of the playing field are labeled as $A, B, C, D$ and the positions of the runners as $P_{1}$ (for the FDJ member) and $P_{2}$ (for the Pioneer) as indicated in the figure. Then the following applies:

$A B=C D=P_{1} P_{2}=70 \mathrm{~m} ; A D=B C=50 \mathrm{~m} ; A P_{1}=$ $P_{1} D=B P_{2}=P_{2} C=25 \mathrm{~m}$.

According to the conditions of the problem, a point is exactly one of the desired flag positions if it

1. lies on the circle with radius $r_{1}=50 \mathrm{~m}$ around $P_{1}$,
2. lies on the circle with radius $r_{2}=25 \mathrm{~m}$ around $P_{2}$, and
3. lies within the rectangle $A B C D$.

Since $r_{1}>r_{2}$ and $r_{1}-r_{2}<P_{1} P_{2}<r_{1}+r_{2}$, there are exactly 2 points that both circles have in common. These points are called $F_{1}$ and $F_{2}$.

Since $r_{1}<P_{1} P_{2}$, $F_{1}$ and $F_{2}$ lie on the same side of the line through $B$ and $C$ as $P_{1}$. The semicircle around $P_{2}$ with radius $r_{2}$ lies entirely on the playing field because $B P_{2}=P_{2} C=r_{2}$. Consequently, $F_{1}$ and $F_{2}$ also lie on the playing field.

Thus, there are exactly these 2 points as possibilities to set up the flag according to the conditions of the problem.