© | Dror Bar-Natan: Classes: 2015-16: Math 475 - Problem Solving Seminar: | (30) |
Next: Blackboards for Tuesday March 8
Previous: Blackboards for Wednesday March 3 |

**Reading.** Sections 1.7, 1.8, and 2.6 of Larson's textbook.

**Next Quiz.** Thursday March 10, on this handout and on sections 1.7 and 1.8.

**Problem 1** (Dror's addition to Larson's 1.6.2).
Of all the $n$-gons with a given perimeter, which has the greatest area?

**Problem 2** (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 3** (Larson's 1.7.8). Determine $F(x)$, if for all real $x$ and $y$, $F(x)F(y)-F(xy)=x+y$.

**Problem 4** (Larson's 2.5.11, modified).

- Let $R_n$ denote the number of ways of placing $n$ nonattacking rooks on an $n\times n$ chessboard so that the resulting arrangement is symmetric about a $90^\circ$ clockwise rotation of the board about its centre. Compute $R_n$.
- Let $S_n$ denote the number of ways of placing $n$ nonattacking rooks on an $n\times n$ chessboard so that the resulting arrangement is symmetric about the centre of the board. Compute $S_n$.
- Let $T_n$ denote the number of ways of placing $n$ nonattacking rooks on an $n\times n$ chessboard so that the resulting arrangement is symmetric about both diagonals. Compute $T_n$.

**Problem 5** (Larson's 2.5.13).

- A
*derangement*is a permutation $\sigma\in S_n$ such that for every $i$, $\sigma i\neq i$. Let $g_n$ be the number of derangements in $S_n$. Show that \[ g_1=0,\quad g_2=1,\quad g_n=(n-1)(g_{n-1}+g_{n-2}). \] Hint. A derangement interchanges $1$ with some other element, or not. - Let $f_n$ be the number of permutations in $S_n$ that have exactly one fixed point (namely, exactly one $i$ such that $\sigma i=i$). Show that $|f_n-g_n|=1$.

**Problem 6** (Larson's 1.8.1, modified).

- Let $0<\alpha<\pi$. Show that $\frac{\sin\theta + \sin(\theta+\alpha)}{\cos\theta-\cos(\theta+\alpha)}$ is independent of $\theta$ for $0\leq\theta\leq\alpha$.
- Can you find a geometric interpretation for this fact?

**Problem 7** (Larson's 1.8.3). In the figure on the right, everything is as it seems: $O$ is the centre, $AB$ is a diameter, $CE$ and $BC$ are tangents, and all lines are straight. Show that $BC=CD$.

**Problem 8.** Which of the 17 tilings patterns appears twice in the picture below?