Wednesday, November 16, 2022
(in North America and South America)
Thursday, November 17, 2022
(outside of North American and South America)
©2022 University of Waterloo
Since the area of one square face of the cube is
Since the edge length of the cube is
Thus,
Answer: 64
Since
Since the sum of the measures of the angles in a triangle is
Since
We note that students who know about “exterior angles” could shorten
this solution by noting that, for example,
Additionally, some students may note that since
Answer:
Since Marie-Pascale solves 4 math problems per day and solves 72
problems in total, this takes her
Since Kaeli solves 54 more problems than Marie-Pascale, she solves
Since Kaeli solves
Answer: 7
Solution 1
Since
Since
Since
Thus,
Solution 2
Suppose that
Since
Since
Since
Therefore,
This solution does not show that this is the value of
Answer:
When three numbers
In other words, if
We note that the common difference
Since Viswanathan’s three scores have this property and his score in the
third game is 25, then his scores in the first two games can be written
as
1st game | 2nd game | 3rd game | |
---|---|---|---|
Viswanathan | |||
Magnus |
Since Viswanathan wins either the first game or the second game, his
score in this game must be at least 25. This means that
Since Viswanathan’s scores in these games are both larger than 25,
Magnus’s scores in these games are either 2 more and 2 less than
Viswanathan’s or 2 less and 2 more than Viswanathan’s, depending on the
order in which they won these games.
Suppose that Viswanathan wins the first game and Magnus wins the
second game.
Then Magnus’s score in the first game is 2 less than Viswanathan’s (and
so is
1st game | 2nd game | 3rd game | |
---|---|---|---|
Viswanathan | |||
Magnus |
Since scores of
Here, this means that Magnus’s score in the third game is
Suppose that Magnus wins the first game and Viswanathan wins the
second game.
Then Magnus’s score in the first game is 2 more than Viswanathan’s (and
so is
1st game | 2nd game | 3rd game | |
---|---|---|---|
Viswanathan | |||
Magnus |
Here, Magnus’s score in the third game is
Therefore, Magnus’s score in the third game is 19. We note that this
answer does not depend on
Answer: 19
For the product
Among the integers in the list, there are five categories with respect
to divisibility by 2 and 3:
(A): 3 are not divisible by 2 or 3 (these are 1, 5, 37)
(B): 3 are divisible by 2 but not by 4 or 3 (these are 22, 46, 50)
(C): 1 is divisible by 4 but not by 3 (this is 8)
(D): 4 are divisible by 3 but not by 2 (these are 21, 27, 33, 39)
(E): 1 is divisible by 2 and by 3 (this is 30)
We work through two possibilities: either 30 is chosen or 30 is not
chosen. We do not need to worry about whether 30 is
Case 1: 30 is chosen
In this case, at least one of the remaining integers is even, since 30 includes only 1 factor of 2.
If both of the remaining integers are even, then 2 of the 4 integers
from (B) and (C) are chosen. There are 6 ways to do this: 22 and 46; 22
and 50; 22 and 8; 46 and 50; 46 and 8; 50 and 8.
Therefore, there are 6 ways to choose in this sub-case.
If only one of the remaining integers is even, this even integer can
come from either (B) or (C). There are 4 such integers.
The third integer chosen must be odd, and so is from (A) or (D). There
is a total of 7 integers in these.
Thus, in this sub-case there are
Case 2: 30 is not chosen
In this case, one or two of the three integers chosen must be from
(D), since
If two of the integers are from (D), there are 6 ways of choosing
these 2 integers (21 and 27; 21 and 33; 21 and 39; 27 and 33; 27 and 39;
33 and 39).
The third integer chosen must then be a multiple of 4 in order for
Thus, in this sub-case there are 6 ways to choose.
If one of the integers is from (D), there are 4 ways of choosing that
integer.
The remaining two integers chosen then need to include a multiple of 4
and another even integer, or a multiple of 4 and an odd integer not
divisible by 3, or two even integers that are not multiples of 4. (Can
you see why these are all of the possible cases?)
If these integers are a multiple of 4 and another even integer, the
multiple of 4 must be 8 and the even integer comes from the 3 integers
in (B).
Thus, there are
If the two integers are a multiple of 4 and an odd integer not divisible
by 3, the multiple of 4 must be 8 and the odd integer comes from the 3
integers in (A).
Thus, there are
If the two integers are even and not multiples of 4, we choose 2 of the
3 integers from (B) and there are 3 ways to do this, as we saw
earlier.
Thus, there are
In total, there are
Answer: 76
Since the sum of the entries in the left column is 29, the bottom
left entry is
Since the sum of the entries in the horizontal row is 29, the remaining
two entries have a sum of
This means that the 3 must go in the bottom right box, which means that
the middle box in the right column must contain
The sums of the left and right columns are equal. Thus,
Since the three sums are equal, we have
Since
Thus, the possible values of
Since the three sums are equal, we have
Since
Putting this together,
Since the figure already contains 10 and 18, we cannot have
This means that the possible values of
When
When
Thus, the vertices of
Since
When
When
Thus, the vertices of
Since
Since the area of
Thus,
When
When
When
When
Since
To calculate the area of quadrilateral
Therefore,
When
From
Solution 1
Since 10 minutes is
Since Hieu cycles at 15 km/h and Beatrice walks at 5 km/h, then Hieu
catches up to Beatrice at a rate of
In other words, Hieu closes the
Solution 2
When Beatrice walks at 5 km/h for 10 minutes, she walks a distance of
Suppose that Hieu takes
Since Hieu cycles at 15 km/h, Hieu cycles
Using similar reasoning, Beatrice walks
When Hieu catches up to Beatrice, they have travelled exactly the same
distance, and so
Thus, Hieu catches up to Beatrice after
There are 60 integers in the interval from 0 to 59
inclusive.
This means that there are 60 possible values for
Since
We say that Beatrice and Hieu meet if they are at the same place on the
path at the same time, including possibly starting or ending
together.
To determine the probability that Beatrice and Hieu meet, we count the
number of pairs of values for
Since the path is 2 km long and Beatrice walks at 5 km/h, then Beatrice
takes
Since the path is 2 km long and Hieu cycles at 15 km/h, then Hieu takes
Since Beatrice walks the path in 24 minutes and Hieu cycles the path in
8 minutes, then Beatrice and Hieu will meet on the path whenever
If Hieu starts before Beatrice (that is, if
If Hieu starts at the same time as Beatrice (that is, if
Suppose that Hieu starts after Beatrice (that is, if
We note that Hieu finishes at
Since
Since Hieu starts after Beatrice and finishes before Beatrice, then Hieu
must pass Beatrice along the path, and so they meet.
If Hieu starts exactly 16 minutes after Beatrice (that is, if
If Hieu starts more than 16 minutes after Beatrice (that is, if
So we need to count the number of pairs of integer values for
To do this, we work through the values of
If
If
In general, if
If
If
As
When
Therefore, if
As in (i), there are 3600 possible pairs of values for
If
We are told that the probability is
As in (i), Hieu takes 8 minutes to cycle the length of the path.
Since Beatrice rides a scooter at
Since
Since
Using a similar analysis to that in (i), Hieu and Beatrice meet whenever
This means that we need to determine the range of possible values of
Since
Let
Note that, since
Note also that
As in (i), we count the number of pairs of values for
When
When
Similarly, while
When
As in (i), the number of possible values of
Since the total number of pairs of values for
Expanding, we obtain
Factoring, we obtain
Since
Since
This means that
Thus,