2013/14 AHSMC Part 1

Of the first $100$ positive integers $1$, $2$, …, $100$, the number of those not divisible by $7$ is

The total score of four students in a test is $2013$. Ace scores $1$ point more than Bea, Bea scores $3$ points more than Cec, and Cec scores $2$ points more than Dee. None of their scores is divisible by

Of the following five fractions, the largest one is

Two teams $A$ and $B$ played a soccer game on each of seven days. On each day, the first team to score seven goals won. There were no ties. Over the seven days, $A$ won more games than $B$, but $B$ scored more goals than $A$ overall. The difference in the total numbers of goals scored by $B$ and $A$ is at most

$ABCD$ is a quadrilateral with $AB = 12$ and $CD = 18$. Moreover, $AB$ is parallel to $CD$ and both $\angle ADC$ and $\angle BCD$ are less than $90^\circ$. $P$ and $Q$ are points on side $CD$ such that $AD = AP$ and $BC = BQ$. The length of $PQ$ is

Each of four cows is either normal or mutant. A normal cow has $4$ legs and always lies. A mutant cow has either $3$ or $5$ legs and always tells the truth. When asked how many legs they have among them, their respective responses are $13$, $14$, $15$ and $16$. The total number of legs among these four cows is

Let $a$ and $b$ be positive integers such that $ab < 100$ and $\dfrac{a}{b} > 2$. Denote the minimum possible value of $\dfrac{a}{b}$ by $m$. Then we have

Let $ABCD$ be a quadrilateral with $\angle DAB = \angle CBA = 90^\circ$. Suppose there is a point $P$ on side $AB$ such that $\angle ADP = \angle CDP$ and $\angle BCP = \angle DCP$. If $AD = 8$ and $BC = 18$, the perimeter of the quadrilateral $ABCD$ is

Two bus routes stop at a certain bus stop. The A bus comes in one-hour intervals while the B bus also comes in regular intervals of a different length. When grandma rests on the bench by the bus stop, one A bus and two B buses come by. Later, grandpa rests on the same bench and eight A buses come by. The minimum number of B buses that must have come by during that time is

Suppose that $16^{2013} = a^b$, where $a$ and $b$ are positive integers. The number of possible values of $a$ is

The following statements are made about the integers $a$, $b$, $c$, $d$ and $e$: (i) $ab$ is even and $c$ is odd; (ii) $bc$ is even and $d$ is odd; (iii) $cd$ is even and $e$ is odd; (iv) $de$ is even and $a$ is odd; (v) $ea$ is even and $b$ is odd. The maximum number of these statements which may be correct is

A very small cinema has only one row of five seats numbered $1$ to $5$. Five movie goers arrive one at a time. Each takes a seat not next to any occupied seat if this is possible. If not, then any seat will do. The number of different orders in which the seats may be taken is

Let $f(x) = x^2 + x + 1$. Let $n$ be the positive integer such that $f(n) = f(20)f(21)$. Then the number of distinct prime divisors of $n$ is

The number of pairs $(x, y)$ of integers satisfying the equation $x^2 + y^2 + xy - x + y = 2$ is

A triangle $ABC$ with $AB = 7$, $BC = 8$ and $CA = 10$ has an interior point $P$ such that $\angle APB = \angle BPC = \angle CPA = 120^\circ$. Let $r_1$, $r_2$ and $r_3$ be the radii of the circles passing through the vertices of triangles $PAB$, $PBC$ and $PCA$ respectively. The value of $r_1^2 + r_2^2 + r_3^2$ is

The list $1$, $3$, $4$, $9$, $10$, $12$, $13$, … contains in increasing order all positive integers which can be expressed as sums of one or more distinct integral powers of $3$. The $100$-th number in this list is