Weekly Dose 15 - Maths Olympiad Practice

1. In a bank, Bava, Juan and Suren hold a distinct position of director (D),
manager (M) and Teller (T). The teller, who is the only child in his family,
earns the least. Suren, who is married to Bava’s sister, earns more than
the manager. What position does Juan hold? Give your answer in terms of
D, M or T.
Solution:
T earns the least and is the only child.
Bava has a sister  Bava cannot be T
Suren earns more than manager, Suren cannot be M and T  Suren is D.
Bava cannot be T and D  Bava is M.
Therefore, Juan is ____
Answer: 𝑇

2. Evaluate 22007
− (22006
+ 22005
+ 22004
+ … + 22
+ 2 + 1).
Solution:
22007 − (22006 + 22005 + 22004 + … + 22 + 2 + 1)
= 22007 − 22006 − 22005 − 22004 − … − 22 − 2 − 1
Because 22007 = 2 × 22006, 22007 − 22006 = 22006
22007
− 22006
− 22005
− 22004
− … − 22
− 2 − 1
= 22006
− 22005
− 22004
− … − 22
− 2 − 1
= 22005 − 22004 − … − 22 − 2 − 1
= 22004
− … − 22
− 2 − 1
⋮
= 22
− 2 − 1
= 2 − 1
= _____
Answer: 1

3. Mary and Peter are running around a circular track of 400m. Mary’s speed
equals
3
5
of Peter’s. They start running at the same point and the same time, but
in opposite directions. 200 seconds later, they have met four times. How many
m per second does Peter run faster than Mary?
Solution:
Mary and Peter met for the forth time after 200 seconds. They met for the first
time after 200 ÷ 4 = 50 seconds.
During that time, they have covered the track once, which means they have
covered a distance of 400𝑚.
Their combined speed is 400 ÷ 50 = 8m/s
Given Mary’s speed(M) =
3
5
of Peter’s speed(P), 𝑀 =
3
5
𝑃
𝑀 + 𝑃 = 8
3
5
𝑃 + 𝑃 = 8
𝑃 = 5𝑚/𝑠
𝑀 = 3𝑚/𝑠
Answer: Peter is faster than Mary by ___

4. In a regular hexagon ABCDEF, two diagonals, FC and BD, intersect at G.
What is the ratio of the area of△BCG to that of quadrilateral FEDG?
The area of △BCG is
1
12
of the hexagon.
The area of FEDG is
5
12
of the hexagon.
The ratio of area of △BCG : area of FEDG =
1
12
:
5
12
= ___ : ___
Answer: 1 ∶ 5
Solution: