CEMC at Home
Grade 9/10 - Monday, May 11, 2020
Contest Day 2 - Solution
Solutions to the two contest problems are provided below, including a video for the second problem.
2020 Canadian Team Mathematics Contest, Team Problem #9
How many times does the digit 0 appear in the integer equal to 20
10
?
Solution:
By factoring and using exponent rules, we have 20
10
= (2
×
10)
10
= 2
10
×
10
10
.
Therefore, 20
10
= 1024
×
10
10
, which is the integer 1024 followed by ten zeros.
Thus, 20
10
has eleven digits that are 0. That is, 10 zeros at the end and one coming from the 1024
at the beginning.
2020 Canadian Team Mathematics Contest, Individual Problem #7
Twenty-seven unit cubes are each coloured completely black or completely red. The unit cubes are
assembled into a larger cube. If
1
3
of the surface area of the larger cube is red, what is the smallest
number of unit cubes that could have been coloured red?
Solution:
Since
3
√
27 = 3, the dimensions of the larger cube must be 3
×
3
×
3.
Therefore, each side of the larger cube has area 3
×
3 = 9.
A cube has 6 faces, so the total surface of the cube is made up of 9
×
6 = 54 of the 1 by 1 squares
from the faces of the unit cubes.
Since
1
3
of the surface area is red, this means
54
3
= 18 of these unit squares must be red.
The unit cube at the centre of the larger cube has none of its faces showing, the 6 unit cubes in the
centres of the outer faces have exactly 1 face showing, the 12 unit cubes on the edge but not at a
corner have 2 faces showing, one of each of two adjacent sides, and the 8 unit cubes at the corners
each have 3 faces showing.
For any unit cube, there are either 0, 1, 2, or 3 of its faces showing on the surface of the larger cube.
This means at most three faces of any unit cube are on the surface of the larger cube. Thus, there
must be at least 6 cubes painted red in order to have 18 red unit squares on the surface of the larger
cube.
There are 8 unit cubes on the corners, so if we colour exactly 6 unit cubes red and the other 21 black,
then arrange the cubes into a 3
×
3
×
3 cube so that the 6 red unit cubes are at the corners, there
will be exactly 18 of the unit squares on the surface coloured red.
Therefore, the answer is 6.
Video
Visit the following link for an explanation of the solution to the second contest problem:
https://youtu.be/K9ax9uQESME