Download Rico can walk 3 miles in the same amount of time that

(1)
Rico can walk 3 miles in the same amount of time that Donna can walk 2
miles. Rico walks a rate 2 miles per hour faster than Donna. At that rate, what is the
number of miles that Rico walks in 2 hours and 10 minutes?
(2)
The first 102 positive integers are separated into successive groups of three
consecutive integers. These groups are arranged in an array of three columns as shown
1
2
3
6
5
4
7
8
9
below. What is the sum of the numbers in the first column? 12 11 10
13 14 15
..
..
..
.
.
.
102 101 100
(3)
(4)
The cost of holding a concert is the sum of the fixed cost, which is the
same no matter how many people attend, and the variable cost, which depends on the
number of people attending. If the total cost for a concert attended by 1000 people is
$75, 000 and the total cost of a concert attended by 1200 people is $85, 000, what is the
number of dollars in the fixed cost when holding a concert?
84 is 40% of what number?
(5)
The sum of two numbers is 15. One number is doubled and the other is
tripled. The sum of the two new numbers is 39. What is the positive difference between
the original numbers?
(6)
Susie’s parents paid x dollars for their house and Brian’s parents paid
$30,000 less for their house than Susie’s parents paid. How many dollars did Brian’s parents
pay for their house? Express your answer as an algebraic expression in terms of x .
(7)
The formula d = 16t 2 is used to calculate the distance, d, in feet, a free
falling object, starting from rest, will travel in t seconds. How many seconds will it take for
a ball, starting from rest, to free fall from a height of 64 feet to the ground?
(8)
Tidy Painters will paint Taylor’s house for a charge of $30 per hour. It takes
40 hours for their crew to complete the job. How many dollars does the paint job cost?
(9)
Given the proportions
your answer as a common fraction.
a
b
=
3
7
and
c
b
=
9
14 ,
what is the value of ca ? Express
(10)
John computes the sum of the elements of each of the 15 two-element
subsets of {1, 2, 3, 4, 5, 6}. What is the sum of these 15 sums?
(11)
Juan, Carlos and Manu take turns flipping a coin in their respective order.
The first one to flip heads wins. What is the probability that Manu will win? Express your
answer as a common fraction.
(12)
A non-square rectangle has integer dimensions. The number of square units
in its area is numerically equal to the number of units in its perimeter. What is the number
of units in the perimeter of this rectangle?
(13)
Of the eggs produced by salmon, 80% hatch, and of those, 25% survive to
migrate to the ocean. How many eggs are needed to produce 100 salmon that migrate to
the ocean?
(14)
Of the 2400 jellybeans in a jar, 1650 are purple. Lucia then removes 150
purple jellybeans from the jar. What fraction of the jellybeans left in the jar are purple?
Express your answer as a common fraction.
(15)
The lengths of the sides of isoceles △ABC are 3x + 62, 7x + 30, and
5x + 50 feet. What is the least possible number of feet in the perimeter of △ABC?
(16)
(17)
Express the reciprocal of 2.3 as a common fraction.
The rectangular region bounded by the lines with equations x = 1.2,
x = 2.6, y = −0.2 and y = d has area 14 square units. What is the greatest possible value
of d? Express your answer as a decimal to the nearest tenth.
(18)
Think of a number n. Double the number. Subtract 160. Divide the result
by 4. Add 60. Subtract half the original number. Now square what you have. What is your
answer?
(19)
Mathman rides
speed in miles per hour?
(20)
Lee can make 18 cookies with two cups of flour. How many cookies can he
make with three cups of flour?
(21)
The Benton Hotel has six floors and the same number of stairs between
floors. It takes Peter 30 seconds to climb the stairs from the first floor to the third floor.
At this rate, how many seconds will it take Peter to climb the stairs from the first floor to
the sixth floor?
3
5
of a mile in 1 21 minutes on his bicycle. What is his average
(22)
What is the number of units in the distance between (2, 5) and (−6, −1)?
(23)
What fraction of a foot is equal to nine inches? Express your answer as a
common fraction.
x
π
≤ 10?
(24)
What is the sum of all integers x that satisfy −5 ≤
(25)
A number x is 3 larger than its reciprocal. What is the value of x −
(26)
Solve for x :
1
6
+
1
6
+
1
6
+
1
6
=
1 4
?
x
x
24 .
1000
1001 ,
B)
1001
1000 ,
C)
(27)
Which of the following has a value closest to 1: A)
101 2
1 − 2−10 , or D) 100
?
(28)
The operation ∗ is defined for non-zero integers as follows: a ∗ b = 1a + b1 . If
a + b = 9 and a × b = 20, what is the value of a ∗ b? Express your answer as a common
fraction.
What is the greatest possible value of x + y such that x 2 + y 2 = 90 and
(29)
x y = 27.
(30)
The sum of the first twenty-one terms of an arithmetic series is 273. The
fifth term is 7. What is the 49th term?
Copyright MATHCOUNTS Inc. All rights reserved
Answer Sheet
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Answer
13
1751
25,000 dollars
210
3
x − 30, 000 or −30, 000 + x
2 seconds
1200 dollars
3/2
105
1/7
18 units
500 eggs
2
3
232
10/23
9.8
400
24
27 cookies
75 seconds
10
3/4
376
81
16
C
9/20
12
51
Problem ID
2A001
A1A4
11D4
05B41
B355
CBC2
C1A2
C4A2
5B13
AC13
D1B4
4443
D1D2
D432
22C31
405C
A5D3
C1B5
B3A2
CBC1
10D4
C1A5
C303
DCA4
4C14
2AC3
0455
3003
AB001
25B5
Copyright MATHCOUNTS Inc. All rights reserved
Solutions
(1) 13
ID: [2A001]
Let the rate (in mph) at which Rico walks be r and the rate (in mph) at which Donna
walks be d. We have the two equations
2
3
=
r
d
r =d +2
The first equation comes from the fact that the time each of Rico and Donna traveled is
equal to the distance traveled divided by the rate of travel. Substituting the second
equation into the first to eliminate r , we have
2
3
=
d +2
d
⇒ 3d = 2d + 4
⇒d =4
Plugging the value of d into the second equation to find r , we get r = 4 + 2 = 6. Thus,
the number of miles Rico traveled in 2 hours and 10 minutes is 6 · (2 + 10/60) = 13 .
(2) 1751
ID: [A1A4]
We split the numbers in the first column into the two arithmetic sequences 1, 7, . . . , 97,
and 6, 12, . . . , 102. Each arithmetic sequence contains 102/6 = 17 terms.
The sum of an arithmetic series is equal to the average of the first and last term,
multiplied by the number of terms, so the sum of the terms in the first series is
(1 + 97)/2 · 17 = 833, and the sum of the terms in the second series is
(6 + 102)/2 · 17 = 918. Therefore, the total sum is 833 + 918 = 1751 .
(3) 25,000 dollars
ID: [11D4]
No solution is available at this time.
(4) 210
ID: [05B41]
No solution is available at this time.
(5) 3
ID: [B355]
No solution is available at this time.
(6) x − 30, 000 or −30, 000 + x
ID: [CBC2]
No solution is available at this time.
(7) 2 seconds
ID: [C1A2]
No solution is available at this time.
(8) 1200 dollars
ID: [C4A2]
No solution is available at this time.
(9) 3/2
ID: [5B13]
No solution is available at this time.
(10) 105
ID: [AC13]
Among the two-element subsets of {1, 2, 3, 4, 5, 6}, each element in {1, 2, 3, 4, 5, 6}
appears 5 times, one time in the same subset
with each other element. Thus, the desired
6·7
sum is 5(1 + 2 + 3 + 4 + 5 + 6) = 5 2 = 105 .
(11) 1/7
ID: [D1B4]
For Manu to win
on his first turn, the sequence of flips would have to be TTH, which has
1 3
probability 2 . For Manu to win on his second turn, the sequence of flips would have to
6
be TTTTTH, which has probability 21 . Continuing, we find that the probability that
3n
Manu wins on his nth turn is 12 . The probability that Manu wins is the sum of these
probabilities, which is
1
1
1
1
23
= 1/7 ,
+
+
+··· =
23 26 29
1 − 213
where we have used the formula a/(1 − r ) for the sum of an infinite geometric series whose
first term is a and whose common ratio is r .
(12) 18 units
ID: [4443]
Let the two sides of the rectangle be a and b. The problem is now telling us ab = 2a + 2b.
Putting everything on one side of the equation, we have, ab − 2a − 2b = 0 This looks
tricky. However, we can add a number to both sides of the equation to make it factor
nicely. 4 works here:
ab − 2a − 2b + 4 = 4 ⇒ (a − 2)(b − 2) = 4
Since we don’t have a square, a and b must be different. It doesn’t matter which one is
which, so we can just say a = 6 and b = 3. The perimeter is then 2(6 + 3) = 18
(13) 500 eggs
ID: [D1D2]
No solution is available at this time.
(14)
2
3
ID: [D432]
No solution is available at this time.
(15) 232
ID: [22C31]
No solution is available at this time.
(16) 10/23
ID: [405C]
No solution is available at this time.
(17) 9.8
ID: [A5D3]
No solution is available at this time.
(18) 400
ID: [C1B5]
No solution is available at this time.
(19) 24
ID: [B3A2]
No solution is available at this time.
(20) 27 cookies
ID: [CBC1]
Let x be the number of cookies that Lee can make with three cups of flour. We can set up
x
the proportion 18
2 = 3 . Solving for x , we find that x = 27 .
(21) 75 seconds
ID: [10D4]
No solution is available at this time.
(22) 10
ID: [C1A5]
p
√
We use the distance formula: (−6 − 2)2 + (−1 − 5)2 = 64 + 36 = 10 .
- OR We note that the points (2, 5), (−6, −1), and (2, −1) form a right triangle with legs of
length 6 and 8. This is a Pythagorean triple, so the length of the hypotenuse must be 10 .
(23) 3/4
ID: [C303]
No solution is available at this time.
(24) 376
ID: [DCA4]
Multiply by π to rewrite the inequality as −5π ≤ x ≤ 10π. Since π ≈ 3.14, we have
−5π ≈ −15.7 and 10π ≈ 31.4. Therefore, the integers satisfying the inequality are those
between −15 and 31 inclusive. The sum of these integers is
−15 − 14 − 13 − · · · + 29 + 30 + 31
= (−15 − 14 − · · · + 14 + 15) + 16 + · · · + 31
= (0) + 16 + · · · + 31
(16 + 31)(16)
2
= 376 ,
=
where we have used the formula
(first term + last term)(number of terms)
2
for the sum of an arithmetic series.
(25) 81
ID: [4C14]
The sentence is telling us, in algebra,
x =3+
1
x
A more useful form for us is
1
=3
x
From there, we can bring both sides to the fourth power:
4
1
= 81
x−
x
x−
(26) 16
ID: [2AC3]
To start, multiply both sides of this equation by 24, to give:
1 1 1 1
24
=x
+ + +
6 6 6 6
Distributing, we find:
24
6
+
24
6
+
24
6
+
24
6
= x , so 4 + 4 + 4 + 4 = x , which means x = 16 .
(27) C
ID: [0455]
No solution is available at this time.
(28) 9/20
ID: [3003]
Note that a ∗ b = 1a +
1
b
=
a+b
ab .
We are given that a + b = 9 and ab = 20. If we substitute
9
these values into a+b
.
ab , we can see that a ∗ b =
20
(29) 12
ID: [AB001]
(x + y )2 = x 2 + y 2 + 2x y = 90 + 2 · 27 = 144, so x + y = 12 or x + y = −12. We want
the larger value, or x + y = 12 .
(30) 51
ID: [25B5]
No solution is available at this time.
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