2023/24 AHSMC Part 1

In a competition, the participants are assigned a positive integer score on a scale from $1$ to $10$ (inclusive). The minimum number of participants required to guarantee that at least three of them achieve the same score is

Alan, Betty, Cece, and Dale are sharing a box of chocolates. They take turns eating one chocolate at a time, and after eating a chocolate, they pass the box to someone among them (other than themselves) who has eaten the fewest chocolates up to that point. After the box is empty, Alan realizes he has eaten four chocolates. The maximum number of chocolates that could have initially been in the box is

In the table, $a$, $b$, $c$, $d$ and $e$ are integers such that every row and column has the same sum. $$\begin{array}{|c|c|c|} \hline a & b & c \\ \hline d & -1 & 2 \\ \hline 4 & e & 3 \\ \hline \end{array}$$ The value of $a$ is

You have two boxes: Box A and Box B. Each box has a square base and is twice as tall as it is wide. Box B is the larger of the two boxes and is two centimeters wider than Box A. If the volume of Box B is $976\,\mathrm{cm}^3$ greater than the volume of Box A, what is the width in cm of Box A?

The number of integers $x$ satisfying $(x^2 - 4)^2 \le (2-x)^2$ is

Oranges cost $99$ cents each, and mangoes cost $\$1.76$ each. You have a $\$13$ gift certificate from the fruit store, and want to use all of it to buy some oranges and mangoes. You want the total cost of the fruit to be as small as possible, provided it is at least $\$13$. What would be the total cost?

What is the number of integers $m$ with $1 \le m \le 2023$ for which $m(m+5)/6$ is an integer?

The number of arrangements of $4$, $5$, $6$, $8$, $9$, $10$, $14$ into a single row so that every pair of adjacent numbers in the arrangement has a common factor greater than $1$ is

Let $f(x) = x^2 + 8x + 13$ with $x$ a real number. The smallest possible value of $f(f(f(f(f(x)))))$ is

An ant is at the point $(0,0)$ and every second it can move one unit either up or right. However, it can only move to the points $(x,y)$ where $x$ and $y$ are not both odd. The number of distinct paths the ant can follow to reach the point $(6,12)$ is

Jamal writes down an increasing sequence of positive integers, starting with $1$, according to the following procedure:

Step $1$: write down the first two numbers, skip one,

Step $2$: write down the next two numbers, skip two,

Step $3$: write down the next two numbers, skip three,

$\ldots$

Step $k$: write down the next two numbers, skip $k$,

$\ldots$

For example, after step $5$, Jamal will so far have written down the numbers $1$, $2$, $4$, $5$, $8$, $9$, $13$, $14$, $19$, $20$. On what step will Jamal first write down a four digit number?

Let $a$, $b$, $c$ be positive integers such that $11a + 8b + 5c = 319$. The smallest possible value of $a+b+c$ is

In a regular polygon $A_1A_2\ldots A_{21}$ the number of triangles $A_i A_j A_k$ having an angle greater than $90^\circ$ is

Let $ABCD$ be a convex quadrilateral whose diagonals $AC$ and $BD$ intersect at point $E$. If $BC = 6$, $DA = 8$ and the triangles $\triangle AEB$ and $\triangle CED$ have the same area, what is the ratio of the area of $\triangle BEC$ over the area of $\triangle DEA$?

The coefficients of a polynomial $P(x)$ are non-negative integers less than $100$. If $P(10) = 234667$ and $P(-10) = 123487$ then the value of $P(1)$ is

In $\triangle ABC$ with $\angle A = 60^\circ$, $D$ is a point on the segment $BC$ such that $\angle BAD = 15^\circ$. If $AB = 4$ and $AD = \sqrt{6}$ then $AC$ is equal to