Weekly Dose 6 - Maths Olympiad Practice

1. Peter has a weigh balance with two pans. He also has one 200g weight
and one 1000g weight. He wants to take 600g of sugar out of a pack
containing 2000g of sugar. What is the minimum number of moves to
accomplish this task?
Solution:
To make it balance, each pan should have
=
𝟐𝟎𝟎𝒈 + 𝟐𝟎𝟎𝟎𝒈 + 𝟏𝟎𝟎𝟎𝒈
𝟐
= 𝟏𝟔𝟎𝟎 𝒈
This also means 𝟔𝟎𝟎𝒈 needs to be moved to the left pan.
Answer: 1 𝑚𝑜𝑣𝑒

2. In the figure below,
𝐸𝐵
𝐵𝐷
=
1
2
and the area of the shaded part is 42cm2. Find
the area of ABC ?
Solution:
Because
𝐸𝐵
𝐵𝐷
=
1
2
, therefore area for
∆𝑩𝑪𝑬 =
1
2
∆𝑩𝑪𝑫
∆𝑨𝑩𝑬 =
1
2
∆𝑨𝑩𝑫
∴ ∆𝑨𝑩𝑪 =
1
2
𝒐𝒇 𝒔𝒉𝒂𝒅𝒆𝒅 𝒂𝒓𝒆𝒂 = ______ 𝒄𝒎 𝟐

3. Solution:
The distance between two adjacent points is:
=
2𝜋 × 2
4
= 𝜋 𝑐𝑚
The distance for every figure ‘8’ travelled = 8𝜋 𝑐𝑚
2005𝜋 ÷ 8𝜋 = 250 𝑟𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 _____
Answer: stop at point _____
In the diagram, there are two touching circles, each of radius 2 cm. An ant
starts at point A and walks around the figure 8 path ABCDEFCGA in that
order. The ant repeats the figure 8 walk, again and again. After the ant has
walked a distance of 2005𝜋 cm it becomes tired and stops. The ant stops
at a point in the path. What letter point is it?

4. Solution:
If A wrote it: A lie, B truth, C lie, D truth, E lie; 3 person lied ⇒ 𝑁𝑜𝑡 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒
If B wrote it: A truth, B lie, C lie, D truth, E lie; 3 person lied ⇒ 𝑁𝑜 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒
If C wrote it: A truth, B truth, C lie, D truth, E lie; 2 person lied ⇒ 𝐴𝑛𝑠𝑤𝑒𝑟
If D wrote it: A lie, B truth, C lie, D truth, E lie; 3 person lied ⇒ 𝑁𝑜 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒
If E wrote it: A lie, B lie, C truth, D lie, E truth; 3 person lied ⇒ 𝑁𝑜 𝑡ℎ𝑒 𝑐𝑎𝑠𝑒
During recess one of the five pupils wrote something nasty on the
blackboard. When questioned by the class teacher, they answered in
following order:
A: “It was B and C.”
B: “Neither E nor I did it.”
C: “A and B are both lying.”
D: “Either A or B is telling the truth.”
E: “D is not telling the truth.”
The class teacher knows that three of them never lie while the other two
may lie. Who wrote it?