2016-2017 Math/Logic Contest
For grades 7 – 8
Solutions
1.
Two truck drivers are driving from city A to city B. The first driver was driving half of the time at a
speed of 40km/h, and half of the time at a speed of 50km/h. The second driver was driving half of
the distance at a speed of 40km/h, and half of the distance at a speed of 50km/h. Which of the
drivers arrived to city B first?
Answer: the first driver will arrive first
Solution. The average speed of the first driver is exactly 45km/h since the driver drove the same
amount of time at the speeds 40km/h and 50km/h. The second driver spent more time driving at a
speed 40km/h since it takes longer to drive the same distance at a lower speed. Therefore, the
average speed of the second driver is less than 45km/h, and the first driver arrived first.
Notes. It is also possible to solve this problem algebraically by, say, introducing variable t representing half
of the total time for the first driver. The time for the second driver can be evaluated in terms of t and then
compared with the time for the first driver. However, it is not enough to consider the case when t = 1h.
2.
Victoria has one 4-litre bucket and one 9-litre bucket. How can Victoria get exactly 6 litres of water
from a river?
Answer: see the chart below
Solution. The steps are described via the following chart:
Steps
4L
9L
1
0
9
2
4
5
3
0
5
4
4
1
5
0
1
6
1
0
7
1
9
8
4
6
Problem is now solved.
Notes. We do not have any information about the geometry of the bucket, and therefore finding half
or third of the bucket volume visually is not possible.
3.
2
The area OBCD is 64m (see the diagram). Point O is the center of the circle. What is the area of the
part of the square OBCD that is outside of the circle?
B
O
Answer: (
C
D
)
2
Solution. Since OBCD is a square with area 64m , the side of the square is equal to 8m.
The radius of the circle shown on the picture is 8m as well, and therefore the total area enclosed by
2
the circle equals to
m . In order to find the area of the part of the square OBCD that is
outside of the circle we subtract the area of the quarter of the circle from the area of the square:
(
4.
Find the missing digits:
8
5
5
5
9
0
Answer: (515950 ÷ 85 = 6070)
)
5.
All positive integer numbers are written together: 1234567891011121314…
What digit is in the 2017th place? (For example, 0 is in the 11th place)
Answer: 7
Solution. There are 9 one-digit numbers, 90 two-digit numbers and 900 three-digit numbers. If written
together, one-digit numbers and two-digit numbers will cover
places. If we add all
three-digit numbers to the list, we will cover
places. This implies that we have a
th
digit of some three-digit number in 2017 place. Now we find this three-digit number.
th
Three-digit numbers start from 190 position (number 100), and each number occupies three positions.
th
In order to get to 2017 position, we need to use
(
)
(
)
th
th
th
three-digit numbers. This means that the first digit of 610 three-digit number is in 2017 place. 610
th
three-digit number is 709, and hence, digit 7 is in the 2017 place.
6.
Each cargo van has a maximum load restriction of 1.5 tons. A client is asking to deliver several boxes
with a total weight of 13.5 tons, and the weight of each box does not exceed 300 kilograms. Show
that 11 cargo vans are sufficient to deliver the 13.5 tons cargo regardless of the weight of each box.
(Answer: Fill the first 10 cargo vans with at least 1.2 tons each [and show why it is possible]. The rest
of boxes weigh 1.5 tons at most, and so they fit in the last cargo van)
Note. Even though each cargo van can carry a load of 1.5 tons , it does not necessarily mean each cargo
van can be fully filled at its maximum capacity. For instance, let us consider scenario when 13.5 tons of
boxes to be delivered are distributed between 51 boxed of weight 260 kg and 1 box of weight 240 kg
(total weight will be 13.5tons). In this particular scenario, it is not possible to fully fill any of the vans.
This is why a simple fact that 11* 1.5 > 13.5 does not necessarily imply that 11 vans are sufficient for
delivery.
Answer: see the algorithm below
Solution. The fact that each box in the cargo does not exceed 300 kg by weight implies that we can fill
each cargo van with at least 1.2 tons. Indeed, we start filling each van with boxes. The moment total
weight in a cargo van exceeds 1.2 tons, we stop filling that cargo van. This means that without the last
box, the total weight in the van is less than 1.2 tons. The resulting weight of the cargo in each van cannot
exceed 1.5 tons since otherwise it would contradict the fact that without the last box, the total weight in
the van is less than 1.2 tons
Now using this fact, we fill the first 10 cargo vans with at least 1.2 tons each. The rest of the boxes weigh
1.5 tons at most [13.5 – 10*1.2], and so they fit in the last cargo van.