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Problem 1

In the expansion of $(a + b)^n$, where $n$ is a natural number, there are $n + 1$ dissimilar terms. Find the number of dissimilar terms in the expansion of $(a + b + c)^n$.

Problem 2

A positive integer $N$ (in base $10$) is called special if the operation $C$ of replacing each digit $d$ of $N$ by its nine’s-complement $9 − d$, followed by the operation $R$ of reversing the order of the digits, results in the original number. (For example, $3456$ is a special number because $R[(C3456)] = 3456$.) Find the sum of all special positive integers less than one million which do not end in zero or nine.

Problem 3

Let a triangle have vertices at $O(0,0)$, $A(a,0)$, and $B(b, c)$ in the $(x, y)$-plane.

  1. Find the coordinates of a point $P(x, y)$ in the exterior of $\triangle OAB$ satisfying $\text{area}(\triangle OAP) = \text{area}(\triangle OBP) = \text{area}(\triangle ABP)$.

  2. Find the coordinates of a point $Q(x, y)$ in the interior of $\triangle OAB$ satisfying $\text{area}(\triangle OAQ) = \text{area}(\triangle OBQ) = \text{area}(\triangle ABQ)$.

Problem 4

A finite set of roads connect $n$ towns $T_1, T_2, …, T_n$ where $n \ge 2$. We say that towns $T_i$ and $T_j$ $(i \ne j)$ are directly connected if there is a road segment connecting $T_i$ and $T_j$ which does not pass through any other town. Let $f(T_k)$ be the number of other towns directly connected to $T_k$. Prove that $f$ is not one-to-one.

Problem 5

Find the function $f(x)$ such that for all $L \ge 0$, the area under the graph of $y = f(x)$ and above the $x$-axis from $x = 0$ to $x = L$ equals the arc length of the graph from $x = 0$ to $x = L$. Hint: recall that $\displaystyle{ \frac{d}{dx}\cosh^{-1} x = \frac{1}{\sqrt{x^2 - 1}} }$.

Problem 6

Let $f(x) = 1/x$ and $g(x) = 1 − x$ for $x \in (0, 1)$. List all distinct functions that can be written in the form $f \circ g \circ f \circ g \circ \cdots \circ f \circ g \circ f$ where $\circ$ represents composition. Write each function in the form $\dfrac{ax + b}{cx + d}$ and prove that your list is exhaustive.

Problem 7

If $a$ and $b$ are real, prove that $x^4 + ax + b = 0$ cannot have only real roots.

Problem 8

A sequence $f_n$ is generated by the recurrence formula

$$ f_{n+1} = \frac{f_n f_{n−1} + 1}{f_{n-2}} $$

for $n = 2, 3, 4, …$, with $f_0 = f_1 = f_2 = 1$. Prove that $f_n$ is integer-valued for all integers $n \ge 0$.