HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #1:
Explanation:
11
Say we want to ascend
elevator, this takes
Solving the inequality
Since
,
floors. By stairs, this take
seconds.
, we get
seconds; by
and
.
should be at least 11.
Answer #2:
Explanation:
30
There are 9 numbers with an arithmetic sequence of difference 0 (1111
through 9999). There are 6 with an arithmetic sequence of difference 1
(1234 through 6789). There are 3 with an arithmetic sequence of difference
2 (1357 through 3579). There are 7 with an arithmetic sequence of
difference −1 (3210 through 9876). There are 4 with an arithmetic
sequence of difference −2 (6420 through 9753), and there is 1 with a
difference of −3 (9630). The answer is therefore 9 + 6 + 3 + 7 + 4 + 1 = 30.
Answer #3:
Explanation:
53/128
Since
, by symmetry, the probability that the total
number of dots is greater than 10 is equivalent to the probability that the
total number is less than 10. The probability that the number is exactly 10 is
( ) ( ) ( ) ( )
,
. Since the sets of dots that total to 10 is
,
,
, and
. Thus the probability that
the total number is greater than 10 is
Answer #4:
Explanation:
365+97/400
(
)
.
HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #5:
Explanation:
38
If a number is almost prime, then it must be the square of a prime. The
three smallest numbers which are squares of primes are , , and ,
for an answer of 4 + 9 + 25 = 38.
Answer #6:
Explanation:
110 degrees
Since the points Y, W, and Q form a straight
line segment, then ∠YWV=180-∠VWQ and so
∠YWV=180-125=55.
Since Q’ is the final position of Q after folding,
then ∠Q’WV is the final position of ∠QWV
after folding, and so ∠Q’WV=∠QWV.
Thus, ∠Q’WV=∠Q’WV=125 and so ∠Q’WY=∠Q’WY-∠YWV=125-55=70.
Since Q’W and R’Y are parallel sides of the piece of paper, then
∠R’YW+∠Q’WY=180, and so ∠R’YW=180-∠Q’WY=180-70=110.
Finally, ∠PYV is opposite ∠R’YW so ∠PYV=∠R’YW=110.
Answer #7: 56
Explanation:
Answer #8:
Explanation:
The number of points on the circle equals the number of spaces between
the points around the circle. Moving from the point labelled 7 to the point
labelled 35 requires moving
points and so 28 spaces around
the circle. Since the points labelled 7 and 35 are diametrically opposite,
then moving along the circle from 7 to 35 results in travelling halfway
around the circle.
Since 28 spaces make half of the circle, then
spaces make the
whole circle. Thus, there are 56 points on the circle, and so
.
113
Clearly, all digits must be odd. 111 does not work as it is divisible by 3, but
113 does because 113, 131, and 311 are all prime.
HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #9:
Explanation:
6
Since the two hexagons are going to be repositioned to
form a square without overlap, the area will remain
the same. The rectangle's area is
. This
means the square will have four sides of length 12. The
only way to do this is shown, therefore
.
Answer #10:
Explanation:
3657421
When the numbers are ordered, the first
numbers all have 1 in
the millions place value. The next 720 numbers all have 2 in the millions
place value. The 2016th number must then lie in the next batch, with 3 as
the millions place value digit. Within the third batch of 720 numbers, the
first
have a 1 in the hundred thousands place value, the next 120
have a 2 in the hundred thousands place, the next 120 have a 4, and so
forth. Continuing in the same manner, we can deduce that the 2016th
number is 3657421.
Answer #11:
Explanation:
8.64
Since 3-4-5 is a Pythagorean triple, the right is a
right triangle. Since the hypotenuse is a
diameter of the circumference, the hypotenuse
is
. Then, the other legs are
and
. the area of the triangle therefore is
.
HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #12:
Explanation:
164
To maximize the sum of the 13 faces that are showing, we can minimize the
sum of the numbers of the 5 faces that are not showing. The bottom 2
cubes each have a pair of opposite faces that are covered up. When the
cube is folded, (1, 32), (2, 16), and (4, 8) are opposite pairs. Clearly 4+8=12
has the smallest sum. The top cube has 1 number that is not showing. The
smallest number on a face is 1. So, the minimum sum of the 5 unexposed
faces is
. Since the sum of the numbers on all the cubes is
(
)
, the maximum possible sum of 13
visible numbers is
.
Answer #13:
Explanation:
1741
Color the diamond layers alternately blue and red, starting
from the outside. You'll get the following pattern:
In the figure Fn, the blue diamonds form a
square, and
) (
) square. Hence
the red diamonds form a (
the total number of diamonds in F30 is
Answer #14:
Explanation:
.
198
Clearly N cannot be a one-digit number. If we try two digits, we get
(
) or
, which again has no solution.
For three digits, we have
(
) , or
. If we want
and
,
, there is only
one solution, namely
,
, and
. If we try four digits, we
easily see that
is far too large for anything to come even
close. Thus the only possible value for N is 198.
HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #15:
Explanation:
38.328
To determine the area of the figure, you can connect the centers of the
circles to form an equilateral triangle with a side of length 4. We must find
the area of this triangle to include the figure formed in between the circles.
Since the equilateral triangle has two 30-60-90 triangles inside, we can find
the height and the base of each 30-60-90 triangle from the ratios: √
The height is √ and the base is 2. Multiplying the height and base
together with , we get √ . Since there are two 30-60-90 triangles in the
equilateral triangle, we multiply the area of the 30-60-90 triangle by 2:
√
√ . To find the area of the remaining sectors, which are of
the original circles once we remove the triangle, we know that the sectors
have a central angle of 300 since the equilateral triangle already covered
that area. Since there are 3 pieces gone from the equilateral triangle, we
have, in total, of a circle (with radius 2) gone. Each circle has an area of
, so three circles gives a total area of
. Subtracting the half
circle, we have:
. Summing the areas from
the equilateral triangle and the remaining circle sections gives us:
is equal to 38.32820…
√ , which when using
Answer #16:
Explanation:
43.96
Let the radius of the smallest circle be . We find that the radius of the
largest circle is
and the radius of the second largest circle is
.
Thus,
, which gives
. The radii of the other circles
(
)
are 3 and 2. The sum of their areas is
.
HighFour Mathematics
Category D: Grades 11 – 12
Round 9
Thursday, May 19, 2016
The use of calculator is required.
Answer #17:
Explanation:
1246
All one-digit numbers have no repeating digits, so that gives us 9 numbers.
For a two-digit number to have no repeating digits, the first digit must be
between 1 and 9, while the second digit must not be equal to the first,
giving us
numbers. For a three-digit number to have no
repeating digits, the first digit must be between 1 and 9, the second digit
must not be equal to the first, and the third digit must not be equal to
either of the two, giving us
numbers. For a four-digit
number between 1000 and 1999 to have no repeating digits, the first digit
must be 1, the second digit must not be equal to the first, and so on, giving
us
numbers. Finally, the only numbers between 2000
and 2016 inclusive with no repeating digits are 2013, 2014, 2015, and 2016.
Therefore, the total is
.
Answer #18:
Explanation:
1
Out of the first two integers, it's possible for both to be odd: for example,
. But the next two integers, when added, increase the sum by
15 which is odd, so one of them must be odd and the other must be even:
for example,
. Finally, the next two integers increase the sum
by 16, which is even, so we can have both be odd: for example,
.
Therefore, the minimum number of integers that must be even is 1.
Answer #19:
Explanation:
2.797
By drawing four lines from the intersect of the semicircles to their centers,
we have split the white region into of a circle with radius 1 and two
equilateral triangles with side length 1. This gives the area of the white
√
√
region as
. The area of the shaded region is the area
of the white region subtracted from the area of the large semicircle. This is
equivalent to
(
is equal to 2.797307…
Answer #20:
Explanation:
√
)
√
. Using
, this expression
18
Note that the area of the parallelogram is double the area of triangle ABC.
If we take AC as the base of the triangle, the height is 3, so the area is
. Thus, the area of the parallelogram is 18.