2024/25 AHSMC Part 1

Let $f(x) = 4x^2 - 4x + 1$ and suppose that $f(a) = 0$ and $f(0) = b$ for some real numbers $a$ and $b$. Then the value of $a+b$ is:

$n$ is a positive integer with at least two digits. When we erase the last digit of $n$, the number we obtain is exactly $\dfrac{n}{13}$. How many such $n$'s are there?

The number of integer pairs $(x,y)$ such that $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{4}$ is:

$n$ is a positive integer with the property that $3500 \le n \le 7500$. The digits of $n$ are $3$, $4$, $5$ and $7$ in some order, without repetition, and $n$ is not divisible by $5$. How many such $n$ exist?

In how many ways can you distribute ten different cookies into five identical boxes so that there are two cookies in each box?

Two boxes contain red and green balls. The first box has $7$ red balls and $5$ green balls. In the second box, there are $5$ red balls and an unknown number of green balls. If a ball is randomly selected from each box, the probability of selecting balls of the same colour is $\dfrac{4}{9}$. How many green balls are in the second box?

A line passes through the points $(3,1)$ and $(k, k+8)$, where $k$ is a positive integer. If the slope of the line is an integer, how many possible values can $k$ take?

$A$, $B$, $C$, $F$ are points on a straight line (in this order). $AB$ is the diameter of a circle, and the line $CE$ is tangent to the circle at point $D$, where $D$ is between $C$ and $E$. If $\angle DCF = 2(\angle ADE - \angle BAD)$, find the value of $\angle ADE$.

$a$, $b$ and $c$ are positive real numbers with the property that $2^a = 3^b = 216^c$. Find the value of $\dfrac{ab}{bc+ca}$.

What is the probability that the equation $x^2 + ax + b = 0$ has exactly one root, when $a$ and $b$ are (not necessarily different) numbers chosen randomly from the set $\{1,2,3,\ldots,100\}$?

You have two rectangular boxes with integer length sides (in cm), one with length $\ell_1$, width $w_1$ and height $h_1$, and the other with length $\ell_2$, width $w_2$ and height $h_2$. As is customary, $\ell_1 \ge w_1$ and $\ell_2 \ge w_2$. Also, $\ell_2 = 3\ell_1$, $w_1 = 2w_2$ and $h_1 = 4 + h_2$. Each box has volume $360$ cubic cm. How many possible values does $\ell_2$ have?

Let $f(x) = x^2 + nx$, where $n$ is an integer. The number of values of $n$ for which the equations $f(x) = 0$ and $f(f(x)) = 0$ have the same set of real solutions (not considering multiplicities) is:

The number of points $(x,y)$ with integer coordinates $x$ and $y$, which are inside the circle $x^2 + y^2 = 1000$, and such that $|3x+1| + |x-1| = y$ is:

How many positive integers $n \le 2024$ have the property that $\log_3(n)$ can be written as the sum of an odd positive integer and some number $0 \le x < 1$?

$m$, $n$ are positive integers with $6 \le n \le 99$. Subject to this condition, $\dfrac{m}{n}$ is the reduced fraction that is the closest possible to $\dfrac{2}{5}$. What is the value of $m+n$?

In triangle $ABC$, $D$ is the midpoint of $BC$ and $AD$ is perpendicular to the bisector of angle $B$. The side lengths of the triangle are positive integers, and one of these sides has length $4$. The number of different triangles possible is: