Problem 1
Three pasture fields have areas of $10/3$, $10$ and $24$ acres, respectively. The fields initially are covered with grass of the same thickness and new grass grows on each at the same rate per acre. If $12$ cows eat the first field bare in $4$ weeks and $21$ cows eat the second field bare in $9$ weeks, how many cows will eat the third field bare in $18$ weeks? Assume that all cows eat at the same rate. (From Math Can be Fun by Ya Perelman.)
Problem 2
A person is engaged in working a jigsaw puzzle that contains $1000$ pieces. It is found that it takes $3$ minutes to put the first two pieces together and that when $x$ pieces have been connected it takes $\dfrac{3(1000 - x)}{1000 + x}$ minutes to connect the next piece. Determine an accurate estimate of the time it takes to complete the puzzle. Give both a formula and an approximate numerical value in hours. (You may find useful the approximate value $\ln 2 = .69$.)
Problem 3
Let $f$ be defined on the natural numbers as follows: $f(1) = 1$ and for $n > 1$, $$f(n) = f(f(n - 1)) + f(n - f(n - 1))$$
Find, with proof, a simple explicit expression for $f(n)$ which is valid for all $n = 1, 2, \dots$.
Problem 4
Suppose that $P(x)$ is a polynomial of degree $3$ with integer coefficients and that $P(1) = 0$, $P(2) = 0$. Prove that at least one of its four coefficients is equal to or less than $-2$.
Problem 5
Determine all real values of $p$ for which the following series converge.
- $\displaystyle{\sum_{n=1}^{\infty} \left( \sin \frac{1}{n} \right)^p}$
- $\displaystyle{\sum_{n=1}^{\infty} \left| \sin n \right|^p}$
Problem 6
The number of individuals in a certain population (in arbitrary real units) obeys, at discrete time intervals, the equation $$ y_{n+1} = y_n (2 - y_n) \quad\text{for } n = 0, 1, 2, \dots, $$ where $y_0$ is the initial population.
- Find all “steady-state” solutions $y^*$ such that, if $y_0 = y^*$, then $y_n = y^*$ for $n = 1, 2, \dots$.
- Prove that if $y_0$ is any number in $(0, 1)$, then the sequence $\lbrace y_n \rbrace$ converges monotonically to one of the steady-state solutions found in (a).
Problem 7
Let the following conditions be satisfied:
- $f = f(x)$ and $g = g(x)$ are continuous functions on $[0, 1]$,
- there exists a number $a$ such that $0 < f(x) \le a < 1$ on $[0, 1]$,
- there exists a number $u$ such that $\displaystyle{\max_{0 \le x \le 1} (g(x) + u f(x)) = u}$.
Find constants $A$ and $B$ such that $F(x) = \dfrac{Ag(x)}{f(x) + B}$ is a continuous function on $[0, 1]$ satisfying $\displaystyle{\max_{0 \le x \le 1} F(x) = u}$, and prove that your function has the required properties.
Problem 8
Ten points in space, no three of which are collinear, are connected, each one to all the others, by a total of $45$ line segments. The resulting framework $F$ will be “disconnected” into two disjoint nonempty parts by the removal of one point from the interior of each of the $9$ segments emanating from any one vertex of $F$. Prove that $F$ cannot be similarly disconnected by the removal of only $8$ points from the interiors of the $45$ segments.