JMC 2020 Challenge

1. Exactly one of the following five numbers is not prime.

Which is it?

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2. What is the value of $2020 ÷ 20%speech%2020 divided by 20$?

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3. Each of these figures is based on a rectangle whose centre is shown.

How many of the figures have rotational symmetry of order two?

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4. How many centimetres are there in $66.6 %speech%66.6$ metres?

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5. Amrita thinks of a number. She doubles it, adds $9$, divides her answer by $3$ and finally subtracts $1$. She obtains the same number she originally thought of.
What was Amrita’s number?

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6. What is the value of $\frac{6}{12}-\frac{5}{12}+\frac{4}{12}-\frac{3}{12}+\frac{2}{12}-\frac{1}{12}%speech%the formula shown on the screen$?

7. Four different positive integers have a product of $110$.

What is the sum of the four integers?

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8. Wesley has a grid of six cells.

He wants to colour two of the cells black so that the two black cells share a vertex but not a side.

In how many ways can he achieve this?

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9. One half of one third of one quarter of one fifth of a number is $2$. What is the number?

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10. How many of these equations have the solution $x = 12 %speech% x equals 12$ ?

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11. The 3 by 3 grid below shows nine $1 \text{ cm} \times 1 \text{ cm}%speech% one cm by one cm,$ squares and uses $24 \text{ cm}%speech% 24 cm$  of wire.

What length of wire is required for a similar 20 by 20 grid?

12.The diagram shows an equilateral triangle divided into four smaller equilateral triangles.

One of these triangles has itself been divided into four smaller equilateral triangles.

What fraction of the area of the large triangle has been shaded?

13. The mean of four positive integers is $5$. The median of the four integers is $6$. What is the mean of the largest and smallest of the integers?

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14. In the diagram, angle 𝑂𝐿𝑀 is twice as large as angle 𝑃𝑂𝑁.What is the size of angle 𝑂𝐿𝑀?

15. A group of $42$ children all play tennis or football, or both sports.

The same number play tennis as play just football. Twice as many play both tennis and football as play just tennis.

How many of the children play football?

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