IMAS 2021–2022 Junior Division, Round 2

Let $x$ and $y$ be non-zero real numbers such that $\begin{cases}x=y+\frac4x,\\y=x+\frac5y.\end{cases}$ Find the sum of all possible values of $|x-y|$.

In the diagram below, $ABCD$ is a rectangle, where $AB=8$ cm and $BC=4$ cm. Let the angle bisector from $D$ intersect $AB$ at $E$. Find the distance, in cm, from $E$ to $AC$.

Using the set of integers from 1 to 2022 inclusive, at most how many integers can we choose such that the positive difference between any two chosen numbers is not a prime number?

The world of the beetles consists of one entire plane divided into unit squares. Initially, all the beetles are located at squares south of an inner wall and each square is occupied by at most one beetle. In each move, a beetle can jump over another beetle in an adjacent square and land on the square immediately beyond. The beetle that was jumped over is then removed. However, the move is not permitted if the square is already occupied. The jump may be northward, eastward or westward. How can one arrange 20 beetles such that they can reach the shaded square in the diagram below?

Numbers 1, 2, 3 and 4 are to be placed into the circles in the diagram shown, where each circle must be filled with exactly one number. If two circles connected by a line should not contain the same number, find the total number of ways in filling the diagram.

We define: $\boxed2=1\times2\times3$, $\boxed3=2\times3\times4$, $\boxed4=3\times4\times5$, $\boxed5=4\times5\times6$, …. If $\frac1{\boxed6}-\frac1{\boxed7}=\frac1{\boxed7}\times A$, what is the value of $A$?

What is the value of the irreducible fraction $\frac nm$, where $0.17<\frac nm<0.18$ and $m$ is as small as possible?

Define $f(x)=\frac1{\sqrt[3]{x^2+2x+1}+\sqrt[3]{x^2-1}+\sqrt[3]{x^2-2x+1}}$, find the value of $f(1)+f(3)+\ldots+f(2k-1)+f(999)$.

What is the last non-zero digit of $2022!$?

Let $ABC$ be a triangle, where $AB=12$ cm, $AC=16$ cm and $BC=20$ cm. If point $P$ lies on segment $BC$, what is the minimum sum of the circumradii, in cm, of triangles $PAB$ and $PAC$?

As shown in the diagram below, we use some number of pentagons, that is identical to $ABCDE$ to fill the plane below. Suppose $AB$ is 1 cm, what is the side length, in cm, of $DE$? Enter the exact expression using sqrt, for example (sqrt(2)+sqrt(3))/2.

Some unit squares of a $6\times4$ board are coloured gray, as shown in the diagram below. Find the number of ways in choosing four different gray unit squares such that any two unit squares that are chosen are located on different rows and different columns.

In a purchase, the quantity of item X must be not more than twice of quantity of item Y, but not less than one third of quantity of item Y. Moreover, the total quantity of the two products must not be more than 60. Find the total number of combinations of non-zero quantities satisfying these requirements.

Seven integers $a_0,a_1,\ldots,a_6$, where $a_i$ is the number of $i$’s among $a_0,a_1,\ldots,a_6$, for each $0\le i\le6$. Find the value of these numbers.

It is known that positive numbers $x,y,z$ satisfy $\begin{cases}x^2+xy+y^2=5^2,\\y^2+yz+z^2=12^2,\\z^2+xz+x^2=13^2.\end{cases}$ What is the value of $(x+y+z)^2$?