Weekly Dose 9 - Maths Olympiad Practice

1. The different triangular symbols represent different digits from 1 to 9. The
symbols represent the same digits in both examples. Find the two-digit
number represented by ?
Solution:
Rewrite the equations to:
𝑎 𝑏 𝑐 𝑎 𝑐 𝑏
× 𝑑 𝑒 × ? ?
𝑓 𝑔 ℎ 𝑐 𝑓 𝑓 𝑐 𝑒
𝑐 ℎ 𝑑 𝑒 𝑐 ℎ 𝑒 𝑑
3 2 8 3 2 3 3 1 5 6
The rightmost ? × 1 = 6, ? = 6
The leftmost ? × 1 = 3. ? = 3
Answer: 36
From the last digit in 3rd row, we know 𝑐 =
2, 𝑒 = 6
From the right, 𝑐 + 𝑑 = 2 + 𝑑 = 5, 𝑑 = 3
From the left, ℎ + 𝑒 = ℎ + 6 = 3, ℎ = 7Rewrite the equations to:
𝑎 𝑏 2 𝑎 2 𝑏
× 3 6 × ? ?
𝑓 𝑔 7 2 𝑓 𝑓 2 6
2 7 3 6 2 7 6 3
3 2 8 3 2 3 3 1 5 6
From the right, compare 𝑓 + 6 and 𝑓 +
7, 𝑓 = 5
From the left, 𝑔 + 3 + 1 = 8, 𝑔 = 4
the last digit for 6 × 𝑏 + 1 = 7, 𝑏 = 1
6 × 𝑎 = 54, 𝑎 = 9
Rewrite the equations to:
9 1 2 9 2 1
× 3 6 × ? ?
5 4 7 2 5 5 2 6
2 7 3 6 2 7 6 3
3 2 8 3 2 3 3 1 5 6

2. Three digit numbers such as 986, 852 and 741 have digits in decreasing
order. But 342, 551, 622 are not in decreasing order. Each number in the
following sequence is composed of three digits:
100, 101, 102, 103, …, 997, 998, 999
How many three digit numbers in the given sequence have digits in
decreasing order?
Solution:
From 0, 1, 2, … , 8, 9 these 10 digits, pick any three and rearrange it to
form decreasing order.
Combination: 𝐶 10, 3 =
10×9×8
1×2×3
= 120
Answer: 120

3. Solution 1:
The time needed to fry 5 fishes (each fish has two sides) one at a time is
2 × 5 × 6 = 60 minutes.
Since 4 fishes can be fried at the same time, total time required can be reduced
to 60 ÷ 4 = 15 minutes.
Let’s name each side of the fishes: 𝐴, 𝑎, 𝐵, 𝑏, 𝐶, 𝑐, 𝐷, 𝑑, 𝐸, 𝑒. We can use the
following method to fry the fishes in 15 minutes.
First 3-minutes  (𝐴, 𝐵, 𝐶, 𝐷)
Second 3-minutes  (𝐵, 𝐶, 𝐷, 𝐸)
Third 3-minutes  (𝑐, 𝑑, 𝐸, 𝐴)
Forth 3-minutes  (𝑑, 𝑒, 𝑎, 𝑏)
Fifth 3-minutes  (𝑒, 𝑎, 𝑏, 𝑐)
Answer: 15 minutes
It takes 6 minutes to fry each side of a fish in a frying pan. Only 4 fish can
be fried at a time. What is the minimum number of minutes needed to fry 5
fish on both sides.

4. Solution:
For each potatoes collected, Jose need to walk 2 × the distance between
the potato and the basket.
The total distance = 2 × 6 × 1 + 2 + 3 + ⋯ + 16
= 2 × 6 × (
17 × 16
2
)
= ________
𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 =
𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑠𝑝𝑒𝑒𝑑
Answer: 544 seconds
A basket and 16 potatoes are placed in a straight line at equal intervals of 6
meters, with the basket fixed at the end. What is the shortest possible time
for Jose to bring the potatoes one by one into the basket, if he starts from
where the basket is and runs at an average speed of 3 meters per second?