2019/20 AHSMC Part 1

Two circles $C_1$, $C_2$ have radius $6$ cm and $8$ cm, respectively. A third circle's area is the same as the sum of the areas of $C_1$ and $C_2$. The radius of the third circle in cm is

The number of positive integers $b$ such that $\log_b(64)$ is an integer is

The sum of the digits of a positive integer $n$ is $2019$. The smallest number of digits $n$ can have is

The polynomial $P(x) = (2x^3 - 1)^{673}$ can be expanded as $$(2x^3 - 1)^{673} = a_{2019} x^{2019} + a_{2018} x^{2018} + \cdots + a_1 x + a_0.$$ Then, the sum $a_{2019} + \cdots + a_1 + a_0$ is

In how many different ways can you write $2020^{2019}$ as the product of two positive integers $x$ and $y$ with no common prime factor? Here, two products of the same numbers in a different order are considered the same.

Consider $a = 2^{1000}$, $b = 3^{600}$, and $c = 10^{300}$. Which of the following is true?

If $(k+2)(x^2 + y^2) \ge 2(x+y)^2$ holds for all real numbers $x$ and $y$, then the minimum possible value of $k$ is

There are exactly two real values of $a$ such that the line $y = ax - 5$ intersects the parabola $y = x^2 + x + 1$ at exactly one point. If $a_1$ and $a_2$ are these values, then $a_1 \cdot a_2$ is equal to

A domino is a rectangular tile divided into $2$ equally sized squares, each of which contains anywhere from $0$ to $100$ dots. Two dominos $A$, $B$ are considered the same when one of the squares on $A$ contains the same number of dots as one of the squares on $B$, and the other square on $A$ contains the same number of dots as the other square on $B$. How many distinct dominos are there?

A tall cabinet $52$ cm wide is standing $50$ cm away from a table $120$ cm high. The cabinet is tilted until it touches the top of the table, as in the figure.

How far from the floor (in cm) is the corner $C$ of the cabinet? Note that the figure is not drawn to scale.

$A$ and $B$ are positive integers such that (i) $A + B = 2019$, (ii) all the digits of $A$ are odd and all the digits of $B$ are even, and (iii) $A$ and $B$ are as close together as possible with these properties. The largest prime number dividing into $B$ is

Four containers are weighed in pairs giving the values $215$ kg, $235$ kg, $260$ kg, $265$ kg, $290$ kg and $310$ kg. The weight (in kg) of the heaviest container is

Each of the integers $1,2,3,\ldots,9$ is coloured either blue, yellow or red such that any two numbers whose difference is odd do not have the same colour. It is not required that all three colours be used. The number of acceptable possibilities is

$p(x)$ is a sixth degree polynomial such that $p(k) = 2k^2 + 1$ for $k = \pm 1$, $k = \pm 2$, $k = \pm 3$. If $p(0) = -71$ then $p(\sqrt{3})$ is

If $a_0 = 1$, $a_1 = 0$ and $a_{n+2} = 3a_{n+1} - 3a_n$ for all integers $n \ge 0$, then $a_{2019}$ is

In triangle $ABC$, $D$ lies inside the segment $BC$ so that $AD \perp BC$, $BD = 12$, $CD = 2$ and $\angle BAC = 60^\circ$. The area of triangle $ABC$ is