Three students each flip three fair coins. What is the probability that all three students get
the same number of tails? Express your answer as a common fraction.
Twins Taylor and Tyler were born on 05/02/07. This date is referred to as a
sum date
because the sum of the month and day is equal to the two-digit year: 05 + 02 = 07. How
many years in the 21st century will have a sum date in each month during that year?
The area of rectangle WXYZ is 90 cm
2
. P and Q are points on
diagonal WY such that 3(WP + QY) = 2PQ. What is the area
of triangle PQZ?
There are 50 equally spaced points marked on a circle. Sara numbers them clockwise from
1 to 50. Starting at 1, she then draws congruent, connected segments between points
that are 8 spaces apart, moving clockwise from the end of the previous segment. For
example, the first segment is drawn from 1 to 9, and the second from 9 to 17. What is the
sum of the numbers on the endpoints of the 23rd segment drawn?
The five squares of the diagram shown are to be colored orange, yellow,
green, blue and indigo, with exactly one color per square. Two colorings
are the same if one is just a rotation of the other (but not if the diagram
must be flipped over). How many distinct colorings are there?
What is the greatest four-digit palindrome that is divisible by 7 and 8?
Tryouts were held for three positions on the school basketball team: center, guard and
forward. There were 4 players who tried out for center, 10 for guard and 10 for forward.
These numbers include 1 player who tried out for center and guard, 3 who tried out for
center and forward, and 4 who tried out for guard and forward. All six of these counts
include 1 player who tried out for all three positions. If 17 players, in all, tried out for the
team, how many players tried out only for guard?
When ice melts and becomes water, its volume decreases by 8%. A cylindrical block of
ice completely fills a container with a height of 10 cm and a radius of 4 cm. When all of the
ice melts, what will be the height of the water in the cylinder? Express your answer as a
decimal to the nearest tenth.
A triangle exists with side lengths 2
x
, 3
x
+ 7 and 6
x
– 5 for how many integer values of
x?
If
x
and
y
are integers, such that
x
>
y
, (
x
+
y
)
2
= 9 and
x
2
+
y
2
= 29, what is the smallest
possible value for
x
?
years
W
P
Q
X
Z
Y
cm
2
colorings
players
cm
values

MATHCOUNTS 2016-2017
30
A circular pizza is cut along four diameters into eight identical sectors. If the total perimeter
of each sector is 10 inches, what is the area of the whole pizza? Express your answer as a
decimal to the nearest tenth.
Because of a traffic jam, Alana’s 18-mile commute to work took 4 minutes longer than
usual, and her average speed was decreased by 9 mi/h. How many minutes did it take her
to get to work that day?
The standard gravitational acceleration of an object near Earth’s surface is
g
n
≈
32 ft/s
2
.

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