2021/22 AHSMC Part 1

Let $x$ be a real number. If $3^{x^4} = 81$, then $x^2$ is

A right triangle is cut out of a square as shown:

The shaded area is equal to

The $25$ little squares of a $5$ by $5$ grid are filled with $1$'s, $2$'s and $3$'s, with each little square getting exactly one number, and so that no two squares touching each other either horizontally or vertically get the same number.

The numbers in the little squares around the outside of the grid are as shown. What is the number in the centre square?

Let $g(k) = 2k^2$ for any positive integer $k$. Let $p \ge 3$ be a prime number. The number of positive integers dividing into $g(g(p))$ is

For how many integers $n$ is the fraction $\dfrac{6300}{n}$ the square of an integer?

Let $A$ be a positive integer so that $25^{303} \cdot 8^{202} = \sqrt{A+1}$. The sum of all digits of the integer $A$ is

In the sequence $1, 4, 1, 4, 4, 1, 4, 4, 4, 1, 4, 4, 4, 4, 1, \ldots$ there are $1$'s separated by strings of $4$'s, where each string of $4$'s is one longer than the previous string. The value of $n$ for which the first $n$ terms sum up to $235$ is

A rectangular block of wood $3$ cm by $4$ cm by $5$ cm is sitting on one of its sides on a table. It is then cut into two pieces by a vertical cut. What (in square cm) is the largest possible sum of the surface areas of the two pieces?

At a daycare consisting of $100$ children, each child can bring at most $10$ toys (but could also bring fewer toys, including no toys at all). Without knowing how many toys each child will bring, what is the maximum value of $n$ for which the following statement is true?

At least $n$ children must bring the same number of toys.

How many $2021$-digit positive integers $n$ satisfy the following three conditions:

• each digit is either $0$ or $1$,
• $n$ is even,
• the number of ones in the decimal representation of $n$ is even.

Shelley is shelling peas. Each pea pod contains either $9$ or $10$ peas. Shelley knows how many peas she has, but she forgot how many pods they came from, and cannot figure it out, even though she is very good at math. What is the smallest number of peas Shelley might have?

Let $a + b + c = 0$, where $a$, $b$, $c$ are non-zero real numbers and $\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} = \dfrac{a}{c} + \dfrac{c}{b} + \dfrac{b}{a} + 2021$. What is the value of $\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a}$?

Let $m$, $n$ be positive integers such that $\dfrac{1}{m} + \dfrac{1}{n} = \dfrac{19}{94}$. The value of $m + n$ is equal to

If $f(x) = \dfrac{1}{2}x^2 + ax + b$, where $a$ and $b$ are real numbers such that $|f(x)| \le (x-1)^2$ for any real $x$, then $f(4)$ is equal to

In triangle $ABC$, $AB = 4$, $AC = 2\sqrt{23}$ and the measure of $\angle BAC = 46^\circ$. $D$ is a point on $AC$ such that $AD = \sqrt{23} + 2$ and $M$ is the midpoint of $BC$. The measure of $\angle ADM$ is

The number of subsets of $\{15, 16, 17, \ldots, 57, 58\}$ having the sum of their elements equal to $1558$ is