2007/08 AHSMC Part 1

A positive integer has $1001$ digits all of which are $1$s. When this number is divided by $1001$, the remainder is

Some cats have got into the pigeon loft because the total head count is $34$ but the total leg count is $80$. The number of cats among the pigeons is

In triangle $ABC$, $AB \le 1 \le BC \le 2 \le CA \le 3$. The maximum area of triangle $ABC$ is

The number of ways in which five As and six Bs can be arranged in a row which reads the same backwards and forwards is

Among twenty consecutive integers each at least $9$, the maximum number of them that can be prime is

The non-negative numbers $x$ and $y$ are such that $2x + y = 5$. The sum of the maximum value of $x + y$ and the minimum value of $x + y$ is

We wish to choose some of the positive integers from $1$ to $1000$ inclusive, such that no two differ by $3$ or $5$. The maximum number of positive integers we can choose is

The number of polynomials $p$ with integral coefficients such that $p(9) = 13$ and $p(13) = 20$ is

In the quadrilateral $ABCD$, $AB = CD$, $AD = 2$ and $BC = 6$. $AD$ and $BC$ are parallel lines at a distance $8$ apart. The radius of the smallest circle which can cover $ABCD$ is

The number of pairs $(a, b)$ of positive integers such that all three roots of the cubic equation $x^3 - 10x^2 + ax - b = 0$ are positive integers is

The real numbers $x$ and $y$ are such that $x + \dfrac{2}{y} = \dfrac{8}{3}$ and $y + \dfrac{2}{x} = 3$. The value of $xy$ is

Let $\theta$ be an acute angle such that $\sec^2\theta + \tan^2\theta = 2$. The value of $\csc^2\theta + \cot^2\theta$ is

The diameter $AC$ divides a circle into two semicircular arcs. $B$ is the midpoint of one these arcs, and $D$ is any point on the other arc. If the area of $ABCD$ is $16$ square centimetres, the distance, in centimetres, from $B$ to $AD$ is

Five students took part in a contest consisting of six true-or-false questions. Student $\#i$ gave the answer T to question $\#j$ if and only if $i < j$. The total number of incorrect answers is $8$ or $9$, and there are more incorrect answers of T than incorrect answers of F. The student who has both an incorrect answer of T and an incorrect answer of F is

An integer $n$ is randomly chosen from $10^{99}$ to $10^{100} - 1$ inclusive. The real number $m$ is defined by $m = \dfrac{9n}{5}$. Of the following five numbers, the one closest to the probability that $10^{99} \le m \le 10^{100} - 1$ is

The smallest value of the real number $k$ such that $(x^2 + y^2 + z^2)^2 \le k(x^4 + y^4 + z^4)$ holds for all real numbers $x$, $y$ and $z$ is