2006/07 AHSMC Part 1

The value of $2^4 4^8 8^{16}$ is

The number of non-congruent rectangles with integer sides and area $2006 = 2 \times 17 \times 59$ is

The number of pairs $(m, n)$ of positive integers such that $m^4 + n = 100000001$ is

In a city, all streets run north-south or east-west, dividing the city into squares. $A$, $B$, $C$ and $D$ are four students who live at four street intersections which define a rectangle, with $A$ and $C$ at opposite corners of this rectangle. They all go to the same school, which is at some street intersection within this rectangle. Each goes to school by a most direct route along streets. $A$ travels $10$ blocks, $B$ $20$ blocks and $C$ $50$ blocks. The number of blocks $D$ travels is

When $(1 + x)(1 + x^2)(1 + x^3)(1 + x^4)(1 + x^5)$ is expanded, the coefficient of the term $x^9$ is

On a plane are two points $A$ and $B$ at a distance $5$ apart. The number of straight lines in this plane which are at a distance $2$ from $A$ and at a distance $3$ from $B$ is

A country has three provinces, each province has three cities, each city has three wards and each ward has three electors. In a two-way election, a candidate wins a ward by getting more votes in the ward, wins a city by winning more wards in the city, wins a province by winning more cities in the province, and wins the election by winning more provinces. Only electors may vote, and they must vote. The minimum number of votes needed to guarantee winning the election is

The teacher asked, “What is the largest possible diameter of a circular coin of negligible thickness which may be stored in a rectangular box with inner dimensions $7 \times 8 \times 9$?” Ace said less than $8$, Bea said $8$, Cec said strictly between $8$ and $9$, Dee said $9$ and Eve said more than $9$. The one who was right was

If $a$ and $b$ are two real numbers which satisfy $a + b - ab = 1$ and $a$ is not an integer, then $b$

The non-zero numbers $a$, $b$, $c$, $x$, $y$ and $z$ are such that $\dfrac{x}{a} = \dfrac{y}{b} = \dfrac{z}{c}$. The value of $$\dfrac{xyz(a + b)(b + c)(c + a)}{abc(x + y)(y + z)(z + x)}$$ is

If $k$ is a rational number such that $\sqrt[3]{9\sqrt{3} - 11\sqrt{2}} = \sqrt{3} + k\sqrt{2}$, then the value of $k$ is

All positive integers which can be expressed as a sum of one or more different integer powers of $5$ are written in increasing order. The first three terms of this sequence are $1$, $5$ and $6$. The fiftieth term is

Consider all polynomials whose coefficients are all integers, whose roots include $\dfrac{\sqrt{3}}{2}$ and $\dfrac{\sqrt{2}}{3}$ and whose degree is as small as possible. Among the coefficients of these polynomials, the smallest positive coefficient is

Colleen used a calculator to compute $\dfrac{a + b}{c}$, where $a$, $b$ and $c$ are positive integers. She pressed $a$, $+$, $b$, $/$, $c$ and $=$ in that order, and got the answer $11$. When she pressed $b$, $+$, $a$, $/$, $c$ and $=$ in that order, she was surprised to get a different answer $14$. Then she realized that the calculator performed the division before the addition. So she pressed $($, $a$, $+$, $b$, $)$, $/$, $c$ and $=$ in that order. She finally got the correct answer, which is

The base of a tetrahedron is an equilateral triangle of side $1$. The fourth vertex is at a distance $1$ above the centre of the base. The radius of the sphere which passes through all four vertices of the tetrahedron is

If $p(x)$ is a polynomial of degree $4$ such that $p(-1) = p(1) = 5$ and $p(-2) = p(0) = p(2) = 2$, then the maximum value of $p(x)$ is