Weekly Dose 18 - Maths Olympiad Practice

1. A giraffe lives in a right-angled triangle. The base and the height of the
triangle are 12m and 16m respectively. The area is surrounded by a fence.
The giraffe can eat the grass outside the fence at a maximum distance of
2m. What is the maximum area outside the fence, in which the grass can
be eaten by the giraffe, in m2? (Given 𝜋 is 3.14 and answer to 2 decimal
places.)
Solution:
Using the Pythagoras theorem, the hypotenuse of the enclosure is 122 + 162 = 20
The area which the giraffe can eat consists of three rectangles: 12 × 2, 16 × 2
and 20 × 2, as well as three circular sectors which together form a circle with
radius 2m.
Hence the area eaten = 12 × 2 + 16 × 2 + 20 × 2 + 𝜋 × 22
= ____
Answer: 108.56 𝑚2

2. If 25% of the people who were sitting stand up, and 25% of the people
who were standing sit down, then 70% of the people are standing. How
many percent of the people were standing initially?
Solution:
Let assume at first A people were sitting, and B people were standing.
Let say C people are standing now.
𝐶 = 25% × 𝐴 + 75% × 𝐵 = 70% × (𝐴 + 𝐵)
25% × 𝐴 + 75% × 𝐵 = 70% × 𝐴 + 70% × 𝐵
5% × 𝐵 = 45% × 𝐴
𝐵
𝐴
=
45%
5%
Or I can write it as 𝐵 ∶ 𝐴 = 9 ∶ 1
∴ Percentage of people standing initially =
𝐵
𝐴+𝐵
× 100% = ___
Answer: 90%

3. A sedan of length 3 metres is chasing a truck of length 17 metres. The sedan is
travelling at a constant speed of 110 kilometres per hour, while the truck is
travelling at a constant speed of 100 kilometres per hour. From the moment
when the front of the sedan is level with the back of the truck to the moment
when the front of the truck is level with the back of the sedan, how many
seconds would it take?
Solution:
The difference of speed is 110𝑘𝑚/ℎ – 100𝑘𝑚/ℎ = 10𝑘𝑚/ℎ
=
10× 1000
60 ×60
=
100
36
𝑚/𝑠𝑒𝑐𝑜𝑛𝑑.
The total distance to overtake the truck is
20m
Time taken = 20 ÷
100
36
= ___ 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Answer: 7.2 seconds

4. Solution:
There are three prime numbers. If the sum of their squares is 5070, what is
the product of these three numbers?
𝑎2 + 𝑏2 + 𝑐2 = 5070 where 𝑎, 𝑏 and 𝑐 is prime number
The unit digit of prime numbers are: 1, 2, 3, 5, 7, 9
The unit digit of the square of any prime numbers are 1, 4, 5, 9
To have the sum of the square of three prime numbers to have 0 as the
unit digit, the unit digit of the square of three prime numbers must be 1, 4,
5.
The prime number that have a square with unit digit 4 can only be 2
The prime number that have a square with unit digit 5 can only be 5
5070 = 22
+ 52
+ 𝑐2
𝑐2 = 5041
𝑐 = 71
The product of 2 × 5 × 71 = ____
Answer: 710