JMC 2020 Challenge cont...

16. You are given the sequence of digits "0625", and can insert a decimal point at the beginning, at the end, or at any of the other three positions.
Which of these numbers can you not make?

17. In 1925, Zbigniew Morón published a rectangle that could be dissected into nine different sized squares as shown in the diagram.

The lengths of the sides of these squares are
$1, 4, 7, 8, 9, 10, 14, 15 \text{ and } 18.%speech%1, 4, 7, 8, 9, 10, 14, 15 and 18$

What is the area of Morón’s rectangle?

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18. How many two-digit primes have both their digits non-prime?

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19. In the table shown, the sum of each row is shown to the right of the row and the sum of each column is shown below the column.

What is the value of L?

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20. Edmund makes a cube using eight small cubes.

Samuel uses cubes of the same size as the small cubes to make a cuboid twice as long, three times as wide and four times as high as Edmund’s cube.

How many more cubes does Samuel use than Edmund?

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21. The digits of both the two-digit numbers in the first calculation below have been reversed to give the two-digit numbers in the second calculation.

The answers to the two calculations are the same.

$62\times 13=806%speech%62 times 13 is 806$     $26\times 31=806%speech%26 times 31 is 806$

For which one of the calculations below is the same thing true?

22. Harriet has a square piece of paper.

She folds it in half to form a rectangle and then in half again to form a second rectangle (which is not a square).

The perimeter of the second rectangle is $30 \text{ cm}%speech%30 centimetres$.

What is the area of the original square?

23. There is more than one integer, greater than $1$, which leaves a remainder of $1$ when divided by each of the four smallest primes.

What is the difference between the two smallest such integers?

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24. Susan is attending a talk at her son’s school. There are $8$ rows of $10$ chairs where $54$ parents are sitting.

Susan notices that every parent is either sitting on their own or next to just one other person.

What is the largest possible number of adjacent empty chairs in a single row at that talk?

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25. In the diagram, $P⁢Q⁢R⁢S$, $J⁢Q⁢K$ and $L⁢R⁢K$ are straight lines

What is the size of the angle $JKL$?

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