Shapovalov A.V.

Three frogs on a swamp jumped in turn. Each landed exactly in the middle of the segment between the other two. The length of the second frog's jump is 60 cm. Find the length of the third frog's jump.
Regardless of how the frogs were sitting initially, after the first jump they will be on one straight line, with the first (A) in the middle

![](https://cdn.mathpix.com/cropped/2024_05_06_df0545ffb539121f37feg-32.jpg?height=62&width=57&top_left_y=2027&top_left_x=406)

A B

Now the second frog (B) jumps. It flies a distance to A and then half of this distance, which according to the condition is 60 cm. Therefore, the distance between it and A (as well as between A and B) was 40 cm.

![](https://cdn.mathpix.com/cropped/2024_05_06_df0545ffb539121f37feg-32.jpg?height=120&width=1246&top_left_y=2255&top_left_x=406)

So now the distance between A and B is 40 cm, B is exactly in the middle between them, and it's V's turn to jump. She will fly 20 cm and then half of this distance, which is a total of 30 cm.

## A 10 cm V 10 cm B $\quad 20 \mathrm{~cm}$

## Answer

30 cm.

The "domino" game. A $10 \times 10$ grid board is given. On a turn, it is allowed to cover any two adjacent cells with a domino (a $1 \times 2$ rectangle) such that the dominoes do not overlap. The player who cannot make a move loses.

## Solution

The second player makes moves symmetric to the first player's moves relative to the center of the board, and wins.