2012/13 AHSMC Part 1

Each day, Mr. Sod visited pubs A, B, C and D in that order, always spending $\$35$, $\$12$, $\$40$ and $\$27$ at the respective places. His total expenditure at the pubs, from the beginning of the month up to a certain moment that month, was $\$1061$. Which pub would he be visiting next?

Meeny, Miny and Moe were playing tennis. From the second game on, the one who sat out the preceding game would replace the loser of that game. At the end, Meeny played $17$ games and Miny played $35$ games. How many games did Moe play?

A circle of diameter $r$ is drawn inside a circle of diameter $R$. For which of the following pairs $(r, R)$ is the area of the smaller circle closest to half the area of the larger circle?

A quadratic polynomial $f(x)$ satisfies $f(0) = 1$, $f(1) = 0$ and $f(2) = 3$. What is the value of $f(3)$?

$ABCD$ is a square. $E$ and $F$ are points on the segment $BC$ such that $BE = EF = FC = 4$ cm. The segments $AF$ and $DE$ intersect at $G$. What, in cm$^2$, is the area of triangle $EFG$?

For how many integers $n \ge 2$ is the sum of the first $n$ positive integers a prime number?

In a test, Karla solved four-fifth of the problems and Klaus solved $35$ problems. Half of the problems were solved by both of them. The number of problems solved by neither was a positive one-digit number. What was this number?

What is the largest possible integer $a$ such that exactly three of the following statements are true: $a < 1$, $a > 2$, $a < 3$, $a > 4$ and $a < 5$?

A rectangle with integer length and width in cm has area $70$ cm$^2$. Which of the following, in cm, cannot be the length of the perimeter of the rectangle?

The positive integer $n$ is such that between $n^2 + 1$ and $2n^2$ there are exactly five different perfect squares. How many such $n$ can we find?

$ABCD$ is a rectangle such that $AD - AB = 15$ cm. $PQRS$ is a square inside $ABCD$ whose sides are parallel to those of the rectangle, with $P$ closest to $A$ and $Q$ closest to $B$. The total area of $APSD$ and $BQRC$ is $363$ cm$^2$ while the total area of $APQB$ and $CRSD$ is $1113$ cm$^2$. What, in cm$^2$, is the area of $PQRS$?

Weifeng writes down $28$ consecutive numbers. If both the smallest and the largest number are perfect squares, what is the smallest number she writes down?

If the positive numbers $a$ and $b$ satisfy $$\frac{1}{a^2 + 4b + 4} + \frac{1}{b^2 + 4a + 4} = \frac{1}{8},$$ what is the maximum value of $a + b$?

The incircle of triangle $ABC$ is tangent to $AB$ and $AC$ at $F$ and $E$ respectively. If $BC = 1$, $\angle A = 90^\circ$ and $\angle B \ne \angle C$, what is the distance from the midpoint of $BC$ to $EF$?

At the beginning of the year, there were more robots than androids. On the first day of each month, each robot made $7$ androids and each android made $7$ robots. The next day, each old android would pick a fight with a new android, and they would destroy each other. At the end of the year, there were $46875$ million robots and $15625$ million androids. What was the difference between the numbers of robots and androids at the beginning of the year?

Let $m$ and $n$ be positive integers such that $11$ divides $m + 13n$ and $13$ divides $m + 11n$. What is the minimum value of $m + n$?