Wednesday, April 3, 2019
(in North America and South America)
Thursday, April 4, 2019
(outside of North American and South America)
©2019 University of Waterloo
Solution 1
Since
Solution 2
Since
In this case, the volume of
We note that since
Since
Since
Since
Note that it is indeed true that
Since
Multiplying by
Checking in the original equation we obtain,
Since the radius of the large circle is 2, its area is
Since the radius of each small circle is 1, the area of each small circle is
Since the two small circles are tangent to each other and to the large circle, then their areas do not overlap and are contained entirely within the large circle.
Since the shaded region consists of the part of the large circle that is outside the two small circles, then the shaded area is
Mo starts at 10:00 a.m. and finishes at 11:00 a.m. and so runs for 1 hour.
Mo runs at 6 km/h, and so runs 6 km in 1 hour.
Thus, Kari also runs 6 km.
Since Kari runs at 8 km/h, then Kari runs for
Since Kari finishes at 11:00 a.m., then Kari started at 10:15 a.m.
The equation
Therefore, the line with this equation has slope
Since the two lines are parallel and the line with equation
Thus, the equation of the second line can be re-written as
Since
Michelle’s list consists of 8 numbers and so its average is
The sum of 7 numbers whose average is 20 is
Since the sum of Michelle’s numbers was 168, then Daphne removed the number equal to
Since
Using exponent laws, the following equations are equivalent:
Solution 1
Since
This means that
Using the ratios of side lengths in a
Since
Since
Since the angles in
This means that
Therefore,
Solution 2
Since
Since
This means that each of
Using the ratios of side lengths in a
Since
Similarly,
Since
Finally, this means that
Since the points
Since distances are always non-negative, the following equations are equivalent:
We can check that the points
First, we note that
Next, we note that
From the first of these, we obtain
From the second of these, we obtain
Thus,
(We are not asked to justify why these are the only two solutions.)
From the second equation, we note that
Rearranging this second equation, we obtain
Substituting into the first equation, we obtain
Since
This means that if
Also, if
Therefore, the values of
Since
This comes from the fact that if
We could list these divisors as
Since 2000 has 20 positive divisors, then there are 20 values of
For example, if
Solution 1
The angles in a polygon with
This means that the angles in a pentagon have a sum of
Also, each interior angle in a regular polygon with
Consider the portion of the regular polygon with
This polygon has 6 sides, and so the sum of its 6 angles is
Four of its angles are the original angles from the
The remaining two angles have measures
We are told that
Also, the angles that measure
Thus,
Therefore,
Solution 2
The angles in a polygon with
This means that the angles in a pentagon have a sum of
Also, each interior angle in a regular polygon with
Consider the portion of the regular polygon with
This polygon has 7 sides, and so the sum of its 7 angles is
Four of its angles are the original angles from the
Two of its angles are the angles equal to
Its seventh angle is the reflex angle corresponding to the pentagon’s angle of
Therefore,
Since the lengths of
Since the angles at
Therefore, the slope of the line segment joining
Also, the slope of the line segment joining
Since the product of the slopes of these two line segments is
Using logarithm and exponent laws, we obtain the following equivalent equations:
Note that if
Note that if
If
Therefore,
Let
Thus, the equation
Since
Thus,
Let
Thus, the equations
If
If
Thus,
If
If
If
By the quadratic formula,
Consider a circle lying inside a quadrilateral
Since
Since
Since
Since
Thus,
This means that
Thus,
Since
Since
This means that
If
Therefore, the following equations are equivalent:
By definition,
By definition,
For
For
Since
Since
Thus, the only positive integer
Suppose that
Assume that
We will show that there must be a contradiction, which will lead to the conclusion that
By definition,
Multiplying by
We treat this as a quadratic equation in
Solving for
Re-writing the discriminant, we obtain
If
If
In either case,
This means that the discriminant can be re-written again as
However, every perfect square is either a multiple of 3 or one more than a multiple of 3:
Suppose that
is an integer and consider . The integer
can be written as one of , , , for some integer . These three cases give
and so is either a multiple of 3 or 1 more than a multiple of 3.
We have determined that the discriminant is a perfect square and is 2 more than a multiple of 3. This is a contradiction.
This means that our initial assumption must be incorrect, and so
Solution 1
We find additional pairs of positive integers
Suppose that
This is equivalent to the equations
Then
The equation
The equation
The equation
The equation
The equation
Thus, the pairs
Solution 2
From (a), we know that
Since the function
Consider the equation
By definition,
Thus,
From
As above, we consider the equation
We obtain the equivalent equations
Continuing in a similar manner, we can also find that
Thus, the pairs
Solution 3
Note that
Where do these pairs come from?
We define the Fibonacci sequence
Thus, the Fibonacci sequence begins
The pairs
We note that
We note that
Furthermore, we note that
Continuing in this way,
On her first two turns, Brigitte either chooses two cards of the same colour or two cards of different colours. If she chooses two cards of different colours, then on her third turn, she must choose a card that matches one of the cards that she already has.
Therefore, the game ends on or before Brigitte’s third turn.
Thus, if Amir wins, he wins on his second turn or on his third turn. (He cannot win on his first turn.)
For Amir to win on his second turn, the second card he chooses must match the first card that he chooses.
On this second turn, there will be 5 cards in his hand, of which 1 matches the colour of the first card that he chose.
Therefore, the probability that Amir wins on his second turn is
Note that there is no restriction on the first card that he chooses or the first card that Brigitte chooses.
For Amir to win on his third turn, the following conditions must be true: (i) the colour of the second card that he chooses is different from the colour of the first card that he chooses, (ii) the colour of the second card that Brigitte chooses is different from the colour of the first card that she chooses, and (iii) the colour of the third card that Amir chooses matches the colour of one of the first two cards.
The probability of (i) is
The probability of (ii) is
The probability of (iii) is
Again, the cards that Amir and Brigitte choose on their first turns do not matter.
Thus, the probability that Amir wins on his third turn is
Finally, the probabilty that Amir wins the game is thus
Suppose that, after flipping the first 13 coins, the probability that there is an even number of heads is
Then the probability that there is an odd number of heads is
When the 14th coin is flipped, the probability of heads is
After the 14th coin is flipped, there can be an even number of heads if the first 13 include an even number of heads and the 14th is not heads, or if the first 13 include an odd number of heads and the 14th is heads.
The probability of this is
Therefore,
If
If
This would mean that the probability of getting an even number of heads when the first 13 coins are flipped is
We could repeat the argument above to conclude that either
Continuing in this way, either one of
This last statement is equivalent to saying that the probability of obtaining a head with the first coin is
Therefore, at least one of
For the sum of the two digits printed to be 2, each digit must equal 1.
Thus,
For the sum of the two digits printed to be 12, each digit must equal 6.
Thus,
For the sum of the two digits printed to be 7, the digits must be 1 and 6, or 2 and 5, or 3 and 4, or 4 and 3, or 5 and 2, or 6 and 1.
Thus,
Since
Since
If
If
Therefore, if we can show that either
Suppose that
Since
From this,
Since
If
But we know that
If
But we know that
Therefore, since we cannot have