Putnam 1991

A $2 \times 3$ rectangle has vertices as $(0, 0), (2,0), (0,3),$ and $(2, 3)$. It rotates $90^\circ$ clockwise about the point $(2, 0)$. It then rotates $90^\circ$ clockwise about the point $(5, 0)$, then $90^\circ$ clockwise about the point $(7, 0)$, and finally, $90^\circ$ clockwise about the point $(10, 0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is (1,1).

Let ${\mathbf{A}}$ and ${\mathbf{B}}$ be different $n \times n$ matrices with real entries. If ${\mathbf{A}}^3 = {\mathbf{B}}^3$ and ${\mathbf{A}}^2 {\mathbf{B}} = {\mathbf{B}}^2 {\mathbf{A}}$, can ${\mathbf{A}}^2 + {\mathbf{B}}^2$ be invertible?

Find all real polynomials $p(x)$ of degree $n \geq 2$ for which there exist real numbers $r_1 < r_2 < \cdots < r_n$ such that

1. $p(r_i) = 0, i = 1, 2, \dots, n,$ and

2. $p' \left( \frac{r_i + r_{i+1}}{2} \right) = 0 i = 1, 2, \dots, n-1,$

where $p'(x)$ denotes the derivative of $p(x)$.

Does there exist an infinite sequence of closed discs $D_1, D_2, D_3, \dots$ in the plane, with centers $c_1, c_2, c_3, \dots$, respectively, such that

1. the $c_i$ have no limit point in the finite plane,

2. the sum of the areas of the $D_i$ is finite, and

3. every line in the plane intersects at least one of the $D_i$?

Find the maximum value of $$\int_0^y \sqrt{x^4 + (y-y^2)^2} dx$$ for $0 \leq y \leq 1$.

Let $A(n)$ denote the number of sums of positive integers $$a_1 + a_2 + \cdots + a_r$$ which add up to $n$ with

$$\begin{gathered}a_1 > a_2 + a_3, a_2 > a_3 + a_4, \ldots, \\ a_{r-2} > a_{r-1} + a_r, a_{r-1} > a_r.\end{gathered}$$

Let $B(n)$ denote the number of $b_1 + b_2 + \cdots + b_s$ which add up to $n$, with

1. $b_1 \geq b_2 \geq \dots \geq b_s,$

2. each $b_i$ is in the sequence $1, 2, 4, \dots, g_j, \dots$ defined by $g_1 = 1$, $g_2 = 2$, and $g_j = g_{j-1} + g_{j-2} + 1,$ and

3. if $b_1 = g_k$ then every element in $\{1, 2, 4, \dots, g_k\}$ appears at least once as a $b_i$.

Prove that $A(n) = B(n)$ for each $n \geq 1$.

(For example, $A(7) = 5$ because the relevant sums are $7, 6+1, 5+2, 4+3, 4+2+1,$ and $B(7) = 5$ because the relevant sums are $4+2+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1.$)

For each integer $n \geq 0$, let $S(n) = n - m^2$, where $m$ is the greatest integer with $m^2 \leq n$. Define a sequence $(a_k)_{k=0}^\infty$ by $a_0 = A$ and $a_{k+1} = a_k + S(a_k)$ for $k \geq 0$. For what positive integers $A$ is this sequence eventually constant?

Suppose $f$ and $g$ are non-constant, differentiable, real-valued functions defined on $(-\infty, \infty)$. Furthermore, suppose that for each pair of real numbers $x$ and $y$,

$$\begin{aligned}f(x+y) &= f(x)f(y) - g(x)g(y), \\ g(x+y) &= f(x)g(y) + g(x)f(y).\end{aligned}$$

If $f'(0) = 0$, prove that $(f(x))^2 + (g(x))^2 = 1$ for all $x$.

Does there exist a real number $L$ such that, if $m$ and $n$ are integers greater than $L$, then an $m \times n$ rectangle may be expressed as a union of $4 \times 6$ and $5 \times 7$ rectangles, any two of which intersect at most along their boundaries?

Suppose $p$ is an odd prime. Prove that $$\sum_{j=0}^p \binom{p}{j} \binom{p+j}{j} \equiv 2^p + 1\pmod{p^2}.$$

Let $p$ be an odd prime and let ${\mathbb{Z}}_p$ denote (the field of) integers modulo $p$. How many elements are in the set $$\{x^2: x \in {\mathbb{Z}}_p\} \cap \{y^2 + 1 : y \in {\mathbb{Z}}_p\}?$$

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^x b^{1-x} \leq a \frac{\sinh ux}{\sinh u} + b \frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0 < |u| \leq c$ and for all $x$, $0 < x < 1$. (Note: $\sinh u = (e^u - e^{-u})/2$.)