6. On a windless day, a polar bear found itself on a small piece of ice that had broken off from an iceberg in the middle of still water. Rescuers from a hovering helicopter noted that the animal was walking in a circle with a diameter of 9.5 meters. How surprised they were when they later saw a chain of the bear's footprints on the photograph, and the diameter of this trajectory turned out to be 10 meters. Estimate the mass of the ice floe, assuming the bear's mass is 600 kg.
Answer: 11.4 t. Solution. When observed from the helicopter, the diameter of the trajectory was measured in the reference frame associated with the ground, while the chain of tracks shows the diameter of the trajectory relative to the ice floe. These trajectories do not coincide because the ice floe and the animal move relative to the center of mass.

By the law of conservation of momentum, the center of mass of the system remains at rest. The distance from the bear to the center of mass $r$ is determined by the formula

$$
m r=M(R-r),
$$

where $R$ is the distance from the center of mass of the ice floe to the bear, $m$ is the mass of the animal, and $M$ is the mass of the ice floe.

From this,

$$
M=\frac{m r}{R-r}=\frac{m d}{D-d}
$$

where $D$ and $d$ are the known diameters of the chain of tracks and the bear's trajectory in the reference frame fixed relative to the ground. Substituting the numbers gives the answer 11400 kg.

Answer to variant 192: 9500 kg $=9.5$ t.

Answer to variant 193: 9000 kg $=9$ t.

Answer to variant 194: 7500 kg $=7.5$ t.

## Grading Criteria

A complete and correct solution to any problem was worth 20 points.

An essentially correct solution containing minor flaws (computational errors, minor deficiencies in reasoning) was worth 15 points.

A solution containing correct reasoning and significant progress, but not leading to the correct answer due to substantial errors, was worth 5 points.

The result of the 5 best problems was counted.