### Just Touching

Three semi-circles have a common diameter, each touches the other two and two lie inside the biggest one. What is the radius of the circle that touches all three semi-circles?

### Sangaku

The square ABCD is split into three triangles by the lines BP and CP. Find the radii of the three inscribed circles to these triangles as P moves on AD.

### Rational Round

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

• Find a shape which is formed by the beads after time $t$.
Extension: In a real situation there will a friction force. The coefficient of friction is $\mu$. Suppose it is not dependent on the speed of a bead. Find the shape of beads after time $t$ and describe how it is changing.