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Thursday, February 24, 2022
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Evaluating,
Answer: (B)
The ones digit of 119 is not even, so 119 is not a multiple of
2.
The ones digit of 119 is not 0 or 5, so 119 is not a multiple of
5.
Since
Since
Finally,
Answer: (D)
The fractions
The fractions
This means that the answer must be either
Using a common denominator of
(Alternatively, we could have converted the fractions into decimals and
used their decimal approximations to compare their sizes.)
Answer: (D)
The sequence consists of a pattern of 5 shapes that are
repeated.
The first repetitions of this pattern end on the 5th, 10th, 15th, 20th,
and 25th shapes.
This means that the 22nd shape is the 2nd shape after the 20th shape,
and so is the 2nd shape in the pattern.
Thus, the 22nd shape is
Answer: (A)
The given sum includes 5 terms each equal to
Thus, the given sum is equal to
Answer: (E)
Yihana is walking uphill exactly when the graph is increasing
(that is, when the slope of the segment of the graph is positive).
This is between 0 and 3 minutes and between 8 and 10 minutes, which
correspond to lengths of time of 3 minutes and 2 minutes in these two
cases, for a total of 5 minutes.
Answer: (A)
Solution 1
Since
Therefore, moving from
Since moving around the whole circle corresponds to moving through
Thus,
Solution 2
We join
Since
Therefore,
Answer: (D)
Since the rectangle has positive integer side lengths and an area
of 24, its length and width must be a positive divisor pair of 24.
Therefore, the length and width must be 24 and 1, or 12 and 2, or 8 and
3, or 6 and 4.
Since the perimeter of a rectangle equals 2 times the sum of the length
and width, the possible perimeters are
These all appear as choices, which means that the perimeter of the rectangle cannot be 36, which is (E).
Answer: (E)
Using the definition,
Assuming
Answer: (A)
Solution 1
Since
Since
Therefore,
Thus,
Solution 2 Since
Since
Therefore,
Thus,
Answer: (D)
The store sells 250 g of jellybeans for $7.50, which is 750
cents.
Therefore, 1 g of jellybeans costs
This means that $1.80, which is 180 cents, will buy
Answer: (C)
Starting with and flipping across
.
Now flipping across , which is the resulting position that Paola sees.
Answer: (E)
Since
Therefore, each of the cubes with volume 8 have a height of 2.
This means that the larger cube has a height of
Answer: (E)
Since 100 000 does not include the block of digits 178, each
integer between 10 000 and 100 000 that includes the block of digits 178
has five digits.
Such an integer can be of the form
The leading digit of a five-digit integer has 9 possible values (any
digit from 1 to 9, inclusive) while a later digit in a five-digit
integer has 10 possible values (0 or any digit from 1 to 9,
inclusive).
This means that
there are 100 integers of the form
there are 90 integers of the form
there are 90 integers of the form
In total, there are thus
Answer: (A)
Since
Since
Answer: (C)
Extend
Since quadrilateral
Thus,
The perimeter of
But quadrilateral
Also,
By the Pythagorean Theorem,
Since
Thus, the perimeter of
(We could have left this as
Of the given choices, this is closest to 63.
Answer: (C)
Zebadiah must remove at least 3 shirts.
If he removes 3 shirts, he might remove 2 red shirts and 1 blue
shirt.
If he removes 4 shirts, he might remove 2 red shirts and 2 blue
shirts.
Therefore, if he removes fewer than 5 shirts, it is not guaranteed that
he removes either 3 of the same colour or 3 of different colours.
Suppose that he removes 5 shirts.
If 3 are of the same colour, the requirements are satisfied.
If no 3 of the 5 shirts are of the same colour, then at most 2 are of
each colour. This means that he must remove shirts of 3 colours, since
if he only removed shirts of 2 colours, he would remove at most
In other words, if he removes 5 shirts, it is guaranteed that there are
either 3 of the same colours or shirts of all 3 colours.
Thus, the minimum number is 5.
Answer: (D)
At the beginning of the first day, the box contains 1 black ball
and 1 gold ball.
At the end of the first day, 2 black balls and 1 gold ball are added, so
the box contains 3 black balls and 2 gold balls.
At the end of the second day,
Continuing in this way, we find the following numbers of balls:
End of Day # | Black Balls | Gold Balls |
---|---|---|
2 | 7 | 4 |
3 | ||
4 | ||
5 | ||
6 | ||
7 |
At the end of the 7th day, there are thus
Answer: (E)
The line with equation
Also, the
Therefore, the triangle bounded by the
We want the area of the triangle bounded by the
This means that
The base of this triangle has length
Thus, we want
Since
Alternatively, we could note that if similar triangles have areas in the
ratio
Since the two triangles in question are similar (both are right-angled
and they have equal angles at the point
Answer: (A)
Since
Since
Since
Since
Using a similar analysis, both
Therefore, we can write
From the given equation,
Since
Since
Since
Neither
Similarly,
Therefore, the smallest possible values of
(We can verify by substitution that
Answer: (C)
Since
Since
Since the units digit of
Since
Thus, the possible values of
We check each of these:
1 | 11 | 870 | Not an integer |
11 | 121 | 760 | 38 |
21 | 231 | 650 | Not an integer |
31 | 341 | 540 | 27 |
41 | 451 | 430 | Not an integer |
51 | 561 | 320 | 16 |
61 | 671 | 210 | Not an integer |
71 | 781 | 100 | 5 |
Therefore, the sum of the smallest and largest of the permissible
values of
Answer: 82
Since the shaded regions are equal, then when the unshaded sector in the small circle is shaded, the area of the now fully shaded sector of the larger circle must be equal to the area of the smaller circle.
The smaller circle has radius 1 and so has area
The larger circle has radius 3 and so has area
This means that the area of the shaded sector in the larger circle has
area
This means that
Thus,
Answer: 40
Suppose that Andreas, Boyu, Callista, and Diane choose the
numbers
There are 9 choices for each of
Among the 9 choices, 5 are odd (
If there are
Therefore, we count the number of quadruples
Among the four integer
If 0 of
If 1 of
If 2 of
If 3 of
If 4 of
Therefore, we need to count the number of quadruples
If 0 are even and 4 are odd, there are 5 choices for each of the
parts, and so there are
If 4 are even and 0 are odd, there are 4 choices for each of the parts,
and so there are
If 2 are even and 2 are odd, there are 4 choices for each of the even
parts and 5 choices for each of the odd parts, and 6 pairs of locations
for the even integers (
In total, this means that there are
The sum of the squares of the digits of
Answer: 78
Let
Let
Since the cube has edge length 8, then
Note that
Let
From directly above, the cube looks like this:
When the sun is directly overhead, the shadow of the cube will look
exactly like the area of the “flat” hexagon
To find the area of this figure, we need to know some lengths, but we
have to be careful because not all of the edges in the edges in the
diagram above are “flat”.
We do know that points
Note that flat quadrilateral
In other words, the area of
So we need to calculate the area of equilateral
Let
Since
By the Pythagorean Theorem in
Since this is in the form
Answer: 67
We begin by tracing what happens when
We start with tokens labelled
We remove the first token (1), move 2 tokens along and remove that token
(3), move 2 tokens along and remove that token (5), and continue around
the circle until we remove tokens 335 and 337.
This leaves tokens
On the second pass, we start with tokens labelled
Because the last token was removed on the first pass (337), the first
token is not removed on the second pass, which means that we remove
every other token starting with 4.
This means that the remaining tokens differ by 4, and are
On the third pass, we start with
Because the last token (336) was removed on the previous pass, we remove
every other token starting with 6.
The remaining tokens differ by 8 and are
On the fourth pass, we start with
Because the last token (334) was removed on the previous pass, we remove
every other token starting with 10.
The remaining tokens differ by 16 and are
On the fifth pass, we start with
Because the last token (330) was removed on the previous pass, we remove
every other token starting with 18.
The remaining tokens differ by 32 and are
On the sixth pass, we start with
Because the second last token (306) was removed on the previous pass, we
remove ever other token starting with 2.
This leaves
On the seventh pass, we remove starting with the second token, which
leaves
On the eighth pass, we remove starting with the first token, which
leaves
This tells us that the smallest possible value of
Next, we will show that
Before first pass:
Therefore, when
Finally, we show that if the final token starting with
Suppose that
Before each pass, the remaining tokens differ by a power of 2, since we
start by removing every other token from a list that differs by 1, then
every other token from a list that differs by 2, and so on.
The smallest powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256.
Since 162 is left after the last pass (this will turn out to be the
eighth pass), the remaining tokens must have differed by 128 before the
eighth pass, and thus were 34, 162. (Since
Thus, before the eighth pass, the tokens were 34, 162 and 34 was
removed.
Before the seventh pass, the tokens differed by 64.
Thus, these were 34, 98, 162. We note that the last token was not
removed on this pass, and so the first token is removed on the eighth
pass, as expected.
Also, there cannot be a token numbered
Before the sixth pass, the tokens differed by 32.
Thus, these were 2, 34, 66, 98, 130, 162, 194. The last token cannot
have been 162 since the last token must be removed on this pass so that
the second token (98) is removed on the seventh pass. Thus,
Before the fifth pass, the tokens differed by 16.
Since the first token (2) is removed on the sixth pass, the last token
is not removed on the fifth pass.
This means that the tokens before this pass were
Before the fourth pass, the tokens differed by 8.
Since the second token (18) is removed on the fifth pass, the last token
is removed on the fourth pass.
This means that the tokens before this pass were
Before the third pass, the tokens differed by 4.
Since the second token (10) is removed on the fourth pass, the last
token is removed on the third pass.
This means that the tokens before this pass were
Before the second pass, the tokens differed by 2.
Since the second token (6) is removed on the third pass, the last token
is removed on the second pass.
This means that the tokens before this pass were
Before the first pass, the tokens differed by 1.
Since the second token (4) is removed on the second pass, the last token
is removed on the first pass.
This means that the tokens before this pass were
Therefore, we must have at least 209 tokens for the final token to be
162, and so the smallest possible value of
Answer: 09
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