Weekly Dose 12 - Maths Olympiad Practice

1. A mathematics exam consists of 20 problems. A student gets 5 points for
a correct answer, a deduction of 1 point for an incorrect answer and no
points for a blank answer. Jolie gets 31 points in the exam. What is the
most number of problems she could have answered (including correct and
incorrect answers)?
Solution:
Because the point Jolie gets is 31, she must at least answer 7 problems
correctly. (from 5 × 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 𝑎𝑛𝑠𝑤𝑒𝑟𝑠 ≥ 31 )
If answered 7 correctly,
7 × 5 − 31 = 4 answered wrongly ⇒ total: 7 + 4 = 11 problems answered
If answered 8 correctly,
8 × 5 − 31 = 9 answered wrongly ⇒ total: 8 + 9 = 17 problems answered
If answered 9 correctly,
9 × 5 − 31 = 14 answered wrongly ⇒ total: 9 + 14 = 23 , not possible.
Answer: the most number of problems she could have answered is 17

2. Nelly correctly measures three sides of a rectangle and gets a total of 88
cm. Her brother Raffy correctly measures three sides of the same
rectangle and gets a total of 80 cm. What is the perimeter of the rectangle
in cm?
Solution:
Let’s name the length of the rectangle as 𝑎 and width as 𝑏.
One of the measurement is: 𝑎 + 𝑏 + 𝑎 = 88 −−→ ①
And another one is: 𝑏 + 𝑎 + 𝑏 = 80 −−→ ②
① + ② 3𝑎 + 3𝑏 = 168
𝑎 + 𝑏 = 56
Perimeter = 2𝑎 + 2𝑏
= 2 𝑎 + 𝑏
= 2 × 56
= 112
Answer: 112 𝑐𝑚

3. Joni and Dini work at the same factory. After every nine days of work, Joni gets
one day off. After every six days of work, Dini gets one day off. Today is Joni’s
day off and tomorrow will be Dini’s day off. At least how many days from today
they will have the same day off?
Solution:
Joni’s day off is 10 days a cycle.
Dini’s day off is 7 days a cycle.
Starting from today, as Dini is only off tomorrow, we are seeking:
a multiple of 10 which is 1 more than multiple of 7
Hence the multiple of 7 must ends with ‘9’. The smallest multiple of 7 which ends
with 9 is 49.
Answer: After 50 days both of them have the same day off.
p.s. You can also answer this question by using table.

4. Solution:
The total weight of 14 cheerleaders = 40 × 14 = 560𝑘𝑔
The total weight of 20 people = 540 + 560 = 1100𝑘𝑔
The average weight of 20 people =
1100
20
= ____ 𝑘𝑔
Answer: 55𝑘𝑔
There are 6 basketball players and 14 cheerleaders. The total weight of the
6 basketball players is 540kg. The average weight of the 14 cheerleaders is
40kg. What is the average weight of all 20 people?