Download How many diagonals does a regular hexagon have?

(1)
How many diagonals does a regular hexagon have?
(2)
A basketball player makes her free throws 55% of the time. What is the
probability that she will make both of her next two free throws? Express your answer as a
common fraction.
(3)
Morgan has 3 hockey shirts, 2 football shirts and 7 baseball shirts in her
closet. If she randomly selects one of these shirts, what is the probability that it will not be
a baseball shirt? Express your answer as a common fraction.
(4)
How many distinct positive integers can be represented as the difference of
two numbers in the set {1, 3, 5, 7, 9, 11, 13}?
(5)
What is the value of 993 + 3(992 ) + 3(99) + 1?
(6)
Container I holds 8 red balls and 4 green balls; containers II and III each hold
2 red balls and 4 green balls. A container is selected at random and then a ball is randomly
selected from that container. What is the probability that the ball selected is green?
Express your answer as a common fraction.
(7)
The perimeter of a square lot is lined with trees, and there are three yards
between the centers of adjacent trees. There are eight trees on a side, and a tree is at
each corner. What is the number of yards in the perimeter of the lot?
(8)
Three segments are chosen at random from six segments having lengths of
2, 3, 5, 6, 7 and 10 units. What is the probability that the three segments chosen could
form a triangle? Express your answer as a common fraction.
(9)
Three points are simultaneously and randomly selected from the 3 by 3 grid
of lattice points shown. What is the probability that they are collinear? Express your
answer as a common fraction.
(10)
Using pennies, nickels, dimes and quarters, what is the least number of coins
needed to make 68 cents in change?
(11)
Ralph and Waldo each simultaneous hold out some fingers on one hand.
Ralph always holds out a prime number of fingers; Waldo always holds out an odd number.
What is the probability that the sum of the number of fingers they hold out is even?
Express your answer as a common fraction.
(12)
A set of seven integers has a median of 73, a mode of 79, and a mean of
75. What is the least possible difference between the maximum and minimum values in the
set?
(13)
Moon has five boxes labeled 1, 2, 3, 4 and 5 which are arranged in
increasing order from left to right. She wants to get them into descending order from left
to right. To do this, she will repeatedly switch the order of two adjacent boxes. What is
the fewest number of switches needed to achieve the desired order?
(14)
Evaluate
1
2!+1
+
2
3!−1
+
3
4!+1 .
Express your answer as a common fraction.
(15)
A point having whole-number coordinates is selected at random from the
line 20x + y = 100. What is the probability that the sum of the coordinates is less than
30? Express your answer as a common fraction.
(16)
If digits may not be repeated, how many positive three-digit integers can be
written using the digits 1, 2, 3 and 4?
(17)
Ms. Albertson is randomly selecting the order in which her 25 students will
each present a report next week. Five students will present each day, Monday through
Friday. What is the probability that the shortest student will present his report on
Thursday? Express your answer as a common fraction.
(18)
A school is creating a four-digit student code system. For security reasons,
the code cannot start with an even number. How many codes are possible?
(19)
What is the probability of getting an even number when a fair six-sided die is
rolled? Express your answer as a common fraction.
(20)
A drawer contains 2 brown and 3 gray socks. The socks are taken out of the
drawer one at a time. What is the probability that the fourth sock removed is gray?
Express your answer as a common fraction.
(21)
Between 3:00 p.m. and 4:00 p.m., for what fractional part of the hour does
the digit 2 appear on a 12-hour digital clock that shows hour and minutes? Express your
answer as a common fraction.
′′ ′′
(22)
Three standard dice are tossed. What is the probability that the sum of the
numbers on the tops of the three dice is 17 or greater? Express your answer as a common
fraction.
(23)
The dartboard below has a radius of 6 inches. Each of the concentric circles
has a radius two inches less than the next larger circle. If nine darts land randomly on the
target, how many darts would we expect to land in a non-shaded region?
(24)
There are six teams in a school district competition. Each team plays each
other team once. What is the total number of games played in the competition?
(25)
A diagonal of a polygon is a line containing two non-consecutive vertices.
How many diagonals does a regular decagon have?
(26)
In each of three boxes are four straws with integer length 1 cm through 4
cm, inclusive. A straw is randomly selected from each box. What is the probability that a
triangle can be formed with the three straws chosen? Express your answer as a common
fraction.
(27)
(28)
What is the value of 1013 − 3 · 1012 + 3 · 101 − 1?
Manu and Janani are playing a coin toss game with a fair penny. Manu gets
a point if the penny lands on heads, and Janani gets a point if the penny lands on tails.
The score is Janani 9, Manu 7, in a game to 10 points. What is the probability that Janani
will win the game?
(29)
Track practice lasts for one hour from 2:30-3:30. At a randomly selected
time during track practice, Tania looks at her watch. What is the probability that the
minute and hour hand on her watch form an acute angle? Express your answer as a
common fraction.
(30)
Twelve students are to be divided among Mr. Mirus’s and Ms. Batty’s
classes. No teacher is to have more than 8 students. How many different groups of
students could be in Mr. Mirus’s class?
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
9
121/400
5/12
6 integers
1000000
5/9
84
9/20
2/21
7
2
3
8
10
64
75
1/3
24
1/5
5000
1/2
3/5
1/4
1/54
6 darts
15
35
17/32
1000000
7/8
1/2
4422
Problem ID
ACA3
B5B41
D0D2
0B42
23C3
2C42
42AC
BB55
21B5
0B13
15C3
13B01
3C55
A5B3
3455
A403
5203
A5B41
0D03
20B3
4303
C413
D102
45C3
1CA5
C4B01
B4B5
B54C
C5D3
A3C3
Copyright MATHCOUNTS Inc. All rights reserved
Solutions
(1) 9
ID: [ACA3]
No solution is available at this time.
(2) 121/400
ID: [B5B41]
No solution is available at this time.
(3) 5/12
ID: [D0D2]
There are 3 + 2 + 7 = 12 shirts to choose from. A total of 2 + 3 = 5 of these, all the
hockey and football shirts, are not baseball shirts. So, the probability of not getting a
5
.
baseball shirt is
12
(4) 6 integers
ID: [0B42]
Since all of the integers are odd, the differences between any pair of them is always even.
So, 13 − 1 = 12 is the largest even integer that could be one of the differences. The
smallest positive (even) integer that can be a difference is 2. So, the integers include 2, 4,
6, 8, 10 and 12, for a total of 6 integers.
(5) 1000000
ID: [23C3]
The given expression is the expansion of (99 + 1)3 . In general, the cube (x + y )3 is
(x + y )3 = 1x 3 + 3x 2 y + 3x y 2 + 1y 3 .
The first and last terms in the given expression are cubes and the middle two terms both
have coefficient 3, giving us a clue that this is a cube of a binomial and can be written in
the form
(x + y )3
In this case, x = 99 and y = 1, so our answer is
(99 + 1)3 = 1003 = 1,000,000
(6) 5/9
ID: [2C42]
There are three different possibilities for our first decision, each corresponding to which
4
= 31
container we choose. So, if we choose container I, with 13 probability, we have a 12
1 1
1
probability for a 3 · 3 = 9 probability of getting green from Container I. Similarly for
container II the probability is 13 · 64 = 29 , and the same for container III. So, the total
5
.
probability is 91 + 92 + 92 =
9
(7) 84
ID: [42AC]
No solution is available at this time.
(8) 9/20
ID: [BB55]
6!
There are 63 = 3!(6−3)!
= 20 ways in which we can choose three segments at random.
Now, recall that, to form a triangle, the sum of the lengths any two sides must be greater
than the length of the third side. So, we cannot make any triangle with 2 and 3, for
example, since 2 + 3 = 5. The only possible triangles are:
2, 5, 6
2, 6, 7
3, 5, 6
3, 5, 7
3, 6, 7
5, 6, 7
5, 6, 10
5, 7, 10
6, 7, 10
9
.
Thus, there are nine possible triangles that we can form, so the probability is
20
(9) 2/21
ID: [21B5]
No solution is available at this time.
(10) 7
ID: [0B13]
No solution is available at this time.
(11)
2
3
ID: [15C3]
No solution is available at this time.
(12) 8
ID: [13B01]
No solution is available at this time.
(13) 10
ID: [3C55]
No solution is available at this time.
(14)
64
75
ID: [A5B3]
No solution is available at this time.
(15) 1/3
ID: [3455]
No solution is available at this time.
(16) 24
ID: [A403]
No solution is available at this time.
(17) 1/5
ID: [5203]
No solution is available at this time.
(18) 5000
ID: [A5B41]
There are five numbers that the code could start with (1, 3, 5, 7, or 9), and each digit of
the code thereafter can be any one of ten numbers (0 through 9). Thus, there are
5 · 103 = 5000 possible codes.
(19) 1/2
ID: [0D03]
No solution is available at this time.
(20) 3/5
ID: [20B3]
No solution is available at this time.
(21) 1/4
ID: [4303]
The digit 2 appears for 10 minutes between 3 : 20 and 3 : 30. It also appears for a minute
at each of 3 : 02, 3 : 12, 3 : 32, 3 : 42, and 3 : 52. So it appears for a total of
1
=
10 + 1 + 1 + 1 + 1 + 1 = 15 minutes, or 15
of the hour.
60
4
(22) 1/54
ID: [C413]
For the sum to be 18, all three dice must come up 6. This happens with probability
1
1
63 = 216 .
For the sum to be 17, one die must come up 5 and the other two must be 6. There are
3
3 ways to choose the die that comes up as a 5, so the probability here is 633 = 216
.
Thus the total probability is
3
4
1
1
+
=
=
216 216
216
54
(23) 6 darts
ID: [D102]
The probability for a single dart to land in the non-shaded region is the ratio of the area of
the non-shaded region to the area of the entire dartboard. The area of the entire dartboard
is π · 62 = 36π. The area of the shaded region is the area of the second largest circle minus
the area of the smallest circle, or π · 42 − π · 22 = 12π, so the area of the non-shaded
24π
region is 36π − 12π = 24π. Thus, our ratio is 36π
= 23 . If each dart has a 32 chance of
landing in a non-shaded region and there are 9 darts, then the expected number of darts
that land in a non-shaded region is 9 · 32 = 6 .
(24) 15
ID: [45C3]
The first team plays each of the five other teams, and then it has no more games. The
second team then plays the four other teams (there are only four because the first team is
no longer playing), and then it is also done. There are three teams left for the third team
to play, two for the fourth, and one final game between the fifth and sixth teams. Thus,
5 + 4 + 3 + 2 + 1 = 15 games are played.
(25) 35
ID: [1CA5]
No solution is available at this time.
(26) 17/32
ID: [C4B01]
No solution is available at this time.
(27) 1000000
ID: [B4B5]
The given expression is the expansion of (101 − 1)3 . In general, the expansion of (a − b)3
is equal to
a 3 − 3 · a 2 · b + 3 · a · b2 − b3
In this case, a = 101, b = 1. Thus 1013 − 3 · 1012 + 3 · 101 − 1 = (101 − 1)3 ; we can
easily calculate 1003 = 1000000 .
(28) 7/8
ID: [B54C]
No solution is available at this time.
(29) 1/2
ID: [C5D3]
No solution is available at this time.
(30) 4422
ID: [A3C3]
No solution is available at this time.
Copyright MATHCOUNTS Inc. All rights reserved