2020/21 AHSMC Part 1

If $\dfrac{2^a}{4^b} = 16$, then $a - 2b$ is equal to

How many integers $x$ satisfy the inequality $\dfrac{x^2 - 61}{x^2 - 16} \le 0$?

Two rectangles have the same area. In the first rectangle the ratio of the long edge to the short edge is $3 : 1$ and in the second rectangle these edges are in a ratio of $4 : 3$. Then the ratio of the perimeter of the first rectangle to the perimeter of the second rectangle is

$1a5$ and $6b9$ are two $3$-digit numbers, where $a$ and $b$ are the middle digits of the two numbers. If $6b9 - 1a5 = 454$ and $6b9$ is divisible by $9$, then $a + b$ is equal to

A number which reads the same backwards as forwards is called a palindrome. For example, $13231$ is a palindrome, since backwards it also reads $13231$. How many palindromes are between $10$ and $2020$?

Let $f$ be the function such that $f(x) = ax^9 + bx^5 + cx^3 - 7$, where $a$, $b$, and $c$ are real numbers. If $f(-9) = 9$ then $f(9)$ is equal to

A doctor's office has a row of chairs, two of which are already occupied. Ruby and Ollie come in and want to sit somewhere in the row, in adjacent chairs, that is, beside each other without a gap. Because of social distancing, there must be at least two empty chairs between them and the persons already sitting. How many chairs at least must the row contain so that Ruby and Ollie can always find a place to sit together, regardless of the location of the occupied chairs?

The number of positive integer divisors of $10! = 10 \cdot 9 \cdot 8 \cdots 2 \cdot 1$ that have a remainder of $1$ when divided by $3$ is

For a positive integer $N$, let $S(N)$ denote the sum of the digits of $N$. Consider the sequence $a_1 = 2$, $a_n = \bigl(S(a_{n-1})\bigr)^2 - 1$ for $n \ge 2$. The term $a_{2020}$ is equal to

In how many ways can a team of $10$ competitors be divided into three groups of two and one group of four (the order of the groups and also the order of the competitors in each individual group does not matter)?

For how many integers $1 \le n \le 100$ is $n^n$ a perfect square?

A right triangle has an angle of $60^\circ$ and the hypotenuse has length $2$ cm. An ant walks around the outside of the triangle always maintaining a distance of exactly $1$ cm to the nearest point on the triangle, and ending where it started. The distance (in cm) travelled by the ant is

Jane has $200$ marbles of various colours. She splits them into groups of the same colour, and she observes that each group has a different number of marbles. The maximum number of groups she can have is

The roots of a cubic polynomial $p(x)$ are the squares of the three solutions of the equation $x^3 - 3x + 1 = 0$. If $p(0) = -4$, then $p(4)$ is equal to

The number of integers $x$ between $10$ and $99$ (inclusive) which have the property that the remainder when $x^3$ is divided by $100$ is equal to the cube of the units digit of $x$ is

Three equal smaller circles are all externally tangent to each other and also internally tangent to a larger circle of radius $10$. The radius of one of the smaller circles is