Scoring for this entire paper: on this platform, start at 0; Q1–15 earn 2 points each and Q16–25 earn 4 points each. Incorrect or blank answers earn 0; maximum 70. Official SASMO rules: start with 15 points; Section A (Q1–15) awards +2/−1/0 for correct/incorrect/blank, and Section B (Q16–25) awards +4/0/0; maximum 85.
A rectangle was made of 100 identical squares. If we remove 1 square from each row and add 5 squares to each column, there will still be 100 squares in the new rectangle. How many columns are there in the original rectangle?
The figure below is made of a circle, a square and a regular hexagon. The circle is centred at $O$ and has an area of $36\text{ cm}^2$. The total area of the unshaded regions in the square and hexagon is 3 times the area of the shaded region. What is the combined area (in $\text{cm}^2$) of the square and hexagon? Use the arrangement shown: $O$ is their common vertex, and the shaded parts are the circle sectors inside the two polygons.
Which cube will be formed when the net below is folded? The letters A–E label the five original choices in the picture. Fold so that the printed faces are on the outside.
What is the next number in the sequence $2,9,28,65,126,\ldots$? Use the intended simple pattern; a finite list alone does not determine a unique continuation.
The operator $\otimes$ acts on two integers to give the following outcomes: $$4\otimes8=818,\quad3\otimes6=616,$$ $$6\otimes12=1222,\quad10\otimes12=2022.$$ What is $20\otimes22$? Use the intended pattern: apply a fixed simple operation separately to each input, then concatenate the two decimal results. These examples are a pattern puzzle, not a unique definition on all integers.
An integer is called beprisque if its two neighbouring integers are a perfect square and a prime number, in either order. For example, 8 is a beprisque number. How many 2-digit beprisque numbers are there?
In square $ABCD$, points $E$ and $F$ lie on sides $AD$ and $CD$, respectively, with $3DE=3FC=AB$. Segments $BF$ and $CE$ intersect at $G$. If the square has area $180\text{ cm}^2$, what is the area (in $\text{cm}^2$) of quadrilateral $EDFG$?
What is the sum of the digits of $3006\times\underbrace{111\ldots111}_{100\text{ ones}}$?
Answer format: enter one whole number without units; leading zeros are optional. The source answer sheet uses four digits (for example, 7 is shaded as 0007).
A train travelling at 108 km/h crosses a bridge in 40 seconds and a man standing on the bridge in 24 seconds. What is the length (in metres) of the bridge? Assume constant speed, and measure each crossing from the front entering until the rear has passed.
Answer format: enter one whole number without units; leading zeros are optional. The source answer sheet uses four digits (for example, 7 is shaded as 0007).
What is the sum of the digits of $A=5555555^2-4444444^2$?
Answer format: enter one whole number without units; leading zeros are optional. The source answer sheet uses four digits (for example, 7 is shaded as 0007).
Let the operation $\triangle$ be given by these examples: $$3\triangle3=2+3+4,\qquad5\triangle2=4+5,$$ $$9\triangle6=8+9+10+11+12+13.$$ If $164\triangle x=2022$, find $x$. Interpret $a\triangle b$ as the sum of $b$ consecutive integers starting at $a-1$, with $b$ a positive integer, as intended by the source.
Answer format: enter one whole number without units; leading zeros are optional. The source answer sheet uses four digits (for example, 7 is shaded as 0007).
A machine converts any positive integer into a two-digit code as follows:
1. The first digit is the remainder when the integer is divided by 8.
2. The second digit is the remainder when it is divided by 13.
3. If the remainder on division by 13 is greater than 9, the second digit is just 9.
For example, 17 becomes 14. What is the smallest positive integer that becomes 12?
Answer format: enter one whole number without units; leading zeros are optional. The source answer sheet uses four digits (for example, 7 is shaded as 0007).
