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 © | Dror Bar-Natan: Classes: 2014-15: Math 475 - Problem Solving Seminar: (27) Next: Blackboards for Thursday February 12 Previous: Quiz #5 - Exploit Symmetry

Handout for February 12, "Exploit Symmetry" and "Divide into Cases"

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).

1. Of all the rectangles which can be inscribed in a given circle, which has the greatest area?
2. Maximize $\sin\alpha+\sin\beta+\sin\gamma$, where $\alpha,\beta,\gamma$ are the angles of a triangle.
3. Of all the triangles of a fixed perimeter, which has the greatest area?
4. Of all the parallelepipeds of volume 1, which has the smallest surface area?
5. Of all the $n$-gons which can be inscribed in a given circle, which has the greatest area?
6. 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 $y$, $F(x)F(y)-F(xy)=x+y$.