© | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: | (27) |
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**Reading.** Sections 1.6 and 1.7 of Larson's textbook.

**Next Quiz.** Tuesday February 24, on these two sections.

**Problem 1.** Prove: You cannot colour the points of the plane with just three colours, so that no two points of distance 1 will be coloured with the same colour. What if you had four colours
available?

**Problem 2** (Larson's 1.6.2).

- Of all the rectangles which can be inscribed in a given circle, which has the greatest area?
- Maximize $\sin\alpha+\sin\beta+\sin\gamma$, where $\alpha,\beta,\gamma$ are the angles of a triangle.
- Of all the triangles of a fixed perimeter, which has the greatest area?
- Of all the parallelepipeds of volume 1, which has the smallest surface area?
- Of all the $n$-gons which can be inscribed in a given circle, which has the greatest area?
- Dror adds: of all the $n$-gons with a given perimeter, which has the greatest area?

**Problem 3** (Larson's 1.7.1). Prove that an angle inscribed in a circle is equal to one half the central angle which subtends the same arc, as in the picture on the right.

**Problem 4** (Larson's 1.7.8). Determine $F(x)$, if for all real $x$ and $y$, $F(x)F(y)-F(xy)=x+y$.