Download HighFour Mathematics Round 5 Category D: Grades 11 – 12

Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #1:
Explanation:
Answer #2:
Explanation:
24
After adding all three equations we obtain
(2𝑎𝑎 + 2𝑏𝑏 + 2𝑐𝑐)(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) = 1152
Factoring out 2 gives
2(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) = 1152
(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)(𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐) = 576
This simplifies to (𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐)2 = 576, which gives that 𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐 = ±24,
giving |𝑎𝑎 + 𝑏𝑏 + 𝑐𝑐| = 24.
9
The number of handshakes that occur when everyone shakes hands with
𝑛𝑛(𝑛𝑛−1)
, where 𝑛𝑛 is the number
everyone else can be found by the formula
2
of people shaking hands.
As such, a table can be made showing these values for values of 𝑛𝑛:
𝑛𝑛(𝑛𝑛−1)
Number of people (𝑛𝑛) Number of handshakes (
13
14
15
78
91
105
2
)
The number of people in the party must be 14 as if it were 15 there would
have been over 100 handshakes. Among the 14 people 91 handshakes
occurred, which means Ms. Taylor knew 9 people.
Answer #3:
Explanation:
210
log 𝑎𝑎
Using the change of base formula log 𝑏𝑏 𝑎𝑎 = 𝑐𝑐 to rewrite 𝑓𝑓(𝑥𝑥) in terms of
log𝑐𝑐 𝑏𝑏
logarithms to the base 3. Thus, 𝑓𝑓(𝑥𝑥) = log 3 4 ∙
which then simplifies to 𝑓𝑓(𝑥𝑥) = log 3 𝑥𝑥. Therefore,
20
20
20
𝑘𝑘=1
𝑘𝑘=1
𝑘𝑘=1
log3 5 log3 6
∙
log3 4 log3 5
� 𝑓𝑓(3𝑘𝑘 ) = � log 3 (3𝑘𝑘 ) = � 𝑘𝑘 = 1 + 2 + ⋯ + 20 = 210
∙ …∙
log3 𝑥𝑥
log3 (𝑥𝑥−1)
,
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #4:
Explanation:
2194.28 square units
Let the legs of the right triangle be 𝑚𝑚 and 𝑛𝑛. The hypotenuse then must be
√𝑚𝑚2 + 𝑛𝑛2 . Then
2
2
2
𝑚𝑚 + 𝑛𝑛 + ��𝑚𝑚2 + 𝑛𝑛2 � = 396
𝑚𝑚2 + 𝑛𝑛2 + 𝑚𝑚2 + 𝑛𝑛2 = 396
2(𝑚𝑚2 + 𝑛𝑛2 )2 = 396
𝑚𝑚2 + 𝑛𝑛2 = 198
From the perimeter we have,
𝑚𝑚 + 𝑛𝑛 + �𝑚𝑚2 + 𝑛𝑛2 = 15 + 20√22
𝑚𝑚 + 𝑛𝑛 + √198 = 15 + 20√22
𝑚𝑚 + 𝑛𝑛 + 3√22 = 15 + 20√22
𝑚𝑚 + 𝑛𝑛 = 15 + 17√22
2
Squaring both sides gives (𝑚𝑚 + 𝑛𝑛)2 = �15 + 17√22� . Expanding the left
side we have 𝑚𝑚2 + 2𝑚𝑚𝑚𝑚 + 𝑛𝑛2 = 225 + 510√22 + 6358 . Using that
𝑚𝑚2 + 𝑛𝑛2 = 198 and simplifying the equation gives 2𝑚𝑚𝑚𝑚 = 6385 +
510√22.
As the area of the triangle is
𝑚𝑚𝑚𝑚
2
, dividing both sides of the equation by 4
gives the area of the triangle, which is
6385+510√22
4
≈ 2194.28 square units.
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #5:
Explanation:
40401
The 200 horizontal lines divide the plane into 201 regions. If each vertical
line is considered in turn, each divides the plane into 201 additional
regions. The total number of regions obtained is 2012 = 40401.
Answer #6:
Explanation:
4
Starting with 4, the powers of 4 are 4, 16, 64, 256, and so forth. It is a
sequence of numbers whose terms have units digit which are alternatively
4 and 6. Therefore, the units digit of the sum of the first n terms is 4 if n is
odd and 0 if n is even.
We are asked to find the units digit of the sum of the first 2011 terms. Since
2011 is odd, the answer is 4.
Answer #7:
Explanation:
0.889
Let D be the event that an item is defective, and N be the event that an
item is non-defective. Using the theorem on total probabilities,
𝑃𝑃(𝐷𝐷) = 𝑃𝑃(𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴) ∙ 𝑃𝑃(𝐷𝐷|𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐴𝐴) + 𝑃𝑃(𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵) ∙ 𝑃𝑃(𝐷𝐷|𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝐵𝐵)
Answer #8:
Explanation:
Using the given values, we get 𝑃𝑃(𝐷𝐷) = (0.3)(0.09) + (0.7)(0.12) = 0.111,
which means that the probability that the selected item is non-defective is
1 − 0.111 = 0.889.
144
The sum of the measures of interior angles of a convex nonagon is 1260°. If
𝑑𝑑 denotes the positive common difference between the consecutive terms
of an arithmetic progression of the interior angles, then
9
𝑆𝑆9 = [2(136°) + (9 − 1)𝑑𝑑)] = 1260°
2
Solving this equation for 𝑑𝑑 gives 𝑑𝑑 = 1, which means the largest interior
angle is 136° + 8(1°) = 144°.
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #9:
Explanation:
1.91
Consider the rotated middle square shown in
the figure. It will drop until length DE is 1 inch.
Then, because DEC is a 45-45-90 triangle,
𝐸𝐸𝐸𝐸 =
√2
,
2
1
and 𝐹𝐹𝐹𝐹 = . We know that 𝐵𝐵𝐵𝐵 =
2
√2, so the distance from B to the line is
1
1
𝐵𝐵𝐵𝐵 − 𝐹𝐹𝐹𝐹 + 1 = √2 − + 1 = √2 + ≈ 1.91
2
2
Answer #10:
0
Explanation:
𝑎𝑎 = �72 + �72 + √72 + ⋯ can be written as 𝑎𝑎2 = 72 + 𝑎𝑎. This may be
written as 𝑎𝑎2 − 𝑎𝑎 − 72 = 0 and may be factored as (𝑎𝑎 − 9)(𝑎𝑎 + 8) = 0,
giving 𝑎𝑎 = 9, since 𝑎𝑎 > 0.
Similarly, 𝑏𝑏 = �90 − �90 − √90 − ⋯ can be written as 𝑏𝑏 2 = 90 − 𝑏𝑏 ,
which then may be written as 𝑏𝑏 2 + 𝑏𝑏 − 90 = 0, which can be factored as
(𝑏𝑏 − 9)(𝑏𝑏 + 10) = 0, giving 𝑏𝑏 = 9, since 𝑏𝑏 > 0.
Answer #11:
Explanation:
Therefore, 𝑎𝑎 − 𝑏𝑏 = 9 − 9 = 0
100 meters
The speed of any point on the first train in relation to the second is
1
65 + 55 = 120 km/h, which is 33 m/s. Therefore, the length of the
1
3
second train is 3 × 33 = 100 meters.
3
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #12:
Explanation:
B
Let us raise both to the power of 9900:
99
9900
� √99!�
100
� √100!�
= (99!)100 = 99! (99!)99
9900
= (100!)99 = (100)99 (99!)99
As 99! Is less than 10099 ,
answer.
Answer #13:
Explanation:
100
√100! Is larger, therefore B is the correct
23.31
Let 𝑥𝑥 denote the length of the side of the square. Thus, the diagonal is
𝑥𝑥 + 2. Using the Pythagoras Theorem, we write (𝑥𝑥 + 2)2 = 𝑥𝑥 2 + 𝑥𝑥 2 , which
simplifies to 𝑥𝑥 2 − 4𝑥𝑥 − 4 = 0. Solving this for the positive root gives
2
𝑥𝑥 = 2�√2 + 1�. Therefore the area of the square is �2�√2 + 1�� ≈ 23.31.
Answer #14:
Explanation:
1034
There are exactly 11 numbers from 1 to 100 that are divisible by 9. For the
product of the numbers written on the pieces of paper to be divisible by 9,
at least one of the numbers must be divisible by 9. Therefore, there are two
cases:
1) When exactly one number is divisible by 9
𝐶𝐶(11,1)𝐶𝐶(89,1)
2) When both numbers are divisible by 9
𝐶𝐶(11,2)
Therefore, 𝐶𝐶(11,1)𝐶𝐶(89,1) + 𝐶𝐶(11,2) = 1034.
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #15:
Explanation:
4
Since triangle DCE and triangle ABD share a base,
the ratio of their areas is the ratio of their altitudes.
Draw the altitude from C to DE.
1
𝑂𝑂𝑂𝑂 = 𝑟𝑟, 𝑂𝑂𝑂𝑂 = 𝑟𝑟.
3
Answer #16:
Explanation:
Since angles DCO and DFC are both 90 degree angles, the triangles DCO and
𝐶𝐶𝐶𝐶
𝑂𝑂𝑂𝑂
1
DFC are similar. So the ratio of the two altitudes is =
= , which may
𝐷𝐷𝐷𝐷
𝐷𝐷𝐷𝐷
3
be written as 1:3. Therefore, the value of X:Y is 1 + 3 = 4.
36
We see that side BE, which we know is 1, is also the shorter leg of one of
the four right triangles (which are congruent, I'll not prove this). So, AH = 1.
Then HB = HE + BE = HE + 1, and HE is one of the sides of the square whose
area we want to find. So:
12 + (𝐻𝐻𝐻𝐻 + 1)2 = √50
1 + (𝐻𝐻𝐻𝐻 + 1)2 = 50
(𝐻𝐻𝐻𝐻 + 1)2 = 49
𝐻𝐻𝐻𝐻 + 1 = 7
𝐻𝐻𝐻𝐻 = 6
Therefore, the area of the square is 62 = 36.
2
Round 5
Tuesday, January 26, 2016
HighFour Mathematics
Category D: Grades 11 – 12
The use of calculator is required.
Answer #17:
Explanation:
4 inches
The original volume of the box is 5 × 6 × 12 = 360. The new volume of the
box is 360 + 1080 = 1440. The volume of the new box may be found as
(𝑥𝑥 + 5)(𝑥𝑥 + 6)(𝑥𝑥 + 12) = 1440.
(𝑥𝑥 2 + 11𝑥𝑥 + 30)(𝑥𝑥 + 12) = 1440
𝑥𝑥 3 + 11𝑥𝑥 2 + 30𝑥𝑥 + 12𝑥𝑥 2 + 132𝑥𝑥 + 360 = 1440
𝑥𝑥 3 + 23𝑥𝑥 2 + 162𝑥𝑥 − 1080 = 0
(𝑥𝑥 − 4)(𝑥𝑥 2 + 27𝑥𝑥 + 270) = 0
Therefore, the increase should be 4 inches.
Answer #18:
Explanation:
10
Let the number be 10𝑎𝑎 + 𝑏𝑏 where 𝑎𝑎 and 𝑏𝑏 are the tens and units digits of
the number. So (10𝑎𝑎 + 𝑏𝑏) − (𝑎𝑎 + 𝑏𝑏) = 9𝑎𝑎 must have a units digit of 6.
This is only possible if 9𝑎𝑎 = 36, so 𝑎𝑎 = 4 is the only way this can be true.
So the numbers that have this property are 40, 41, 42, 43, 44, 45, 46, 7, 48,
and 49. Therefore the answer is 10.
Answer #19:
Explanation:
1
To begin, we see that the remaining 30% of the students got 95 points.
Assume that there are 20 students; we see that 2 students got 70 points, 5
students got 80 points, 4 students got 85 points, 3 students got 90 points,
and 6 students got 95 points. The median is 85, since the 10th and 11th
70(2)+80(5)+85(4)+90(3)+95(6)
=
terms are both 85. The mean is
20
The difference between the mean and median, therefore, is 1.
Answer #20:
Explanation:
1720
20
= 86.
26
Since Bertha has 6 daughters, she has 30 − 6 = 24 granddaughters, of
24
which none have daughters. Of Bertha's daughters, = 4 have daughters,
6
so 6 − 4 = 2 do not have daughters. Therefore, of Bertha's daughters and
granddaughters, 24 + 2 = 26 do not have daughters.