MATHCOUNTS

®
KJ School Competition #2 
Sprint Round
Problems 1 – 30
Name _______________________________________________
Grade :__________
Teacher ____________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
This round of the competition consists of 30 problems. You will
have 40 minutes to complete the problems. You are not allowed to
use calculators, books, or any other aids during this round. If you
are wearing a calculator wrist watch, please give it to your proctor
now. Calculations may be done on scratch paper. All answers
must be complete, legible, and simplified to lowest terms. Record
only final answers in the blanks in the right-hand column of the
competition booklet. If you complete the problems before time is
called, use the remaining time to check your answers.
In each written round of the competition, the required unit for the
answer is included in the answer blank. The plural form of the unit
is always used, even if the answer appears to require the singular
form of the unit. The unit provided in the answer blank is the only
form of the answer that will be accepted.
Total Correct
Scorer’s Initials
1.
Brage runs around the track at a
steady rate, finishing 400 meters in
exactly two minutes. At that rate, how
many minutes will it take him to run
4000 meters?
1.
minutes
2.
If
7.
2.
__
3.
In the below diagram two squares have been drawn. Given
that the larger square’s side length is equal to the smaller
square’s diagonal length and that the smaller square has a
side length of 6 cm, find the area of the shaded region.
7.
3.
cm_2
4.
Find the value of
4.
_
5.
How many faces does an
octagonal prism have?
5.
faces
6.
I will choose a random integer, Q, such that
.
What is the probability, expressed as a common fraction,
that Q is prime?
6.
_
7.
40 times a number equals 30 times 1 more than the same
number. Find the number.
7.
__
find
as a decimal to the nearest hundredth.
8.
I roll a die 7 times and get a prime
number on every single roll. What is
the probability I will get another
prime number on my 8th roll. Express
your answer as a common fraction.
8.
_
9.
What is the single discount that is equivalent to the two
consecutive discounts of 10% off followed by 30% off the
discounted price?
9.
%
10. How many perfect squares can be found in the first 2011
positive integers?
10.
squares
11. Find the area of a trapezoid,
in square inches, with bases
of 3 inches and 5 inches and
a height 2 feet.
11.
in2
12. A palindrome is a number that reads the same way
forwards and backwards. How many 2-digit palindromes
are prime?
12.
palindromes
13. I bought a house for $200,000 dollars in 2005. Given that
the house’s value depreciates at a
rate of 8%, in how many years
will the house first have a value
less than $150,000?
13.
years
14.
14.
cm2
15.
_
Find the area of a regular hexagon with
side length 2 cm. Express your answer in
simplest radical form.
15. What is the smallest positive integer, , such that the
product of 60 and is a perfect square?
16. Miles has a 3-digit locker code. If he knows there are no 0’s
in the code and exactly 2 5’s in it, how many codes could
Miles possibly have?
16.
codes
17. How many positive factors does 333 have?
17.
factors


18. In the MATHCOUNTS Summer Camp, exactly of the
18.
boys
19. Find
19.
_
20. What is the measure of the smaller angle,
rounded to the nearest whole number,
formed by the hour and minute hand when
a standard clock reads 8:31?
20.
degrees
21.
_
22.
_
23.
ft2
participants are boys. Exactly
of all participants have
glasses and exactly of all the female participants have
glasses. Find the least number of boys that have glasses.
21. Find the sum of the first 50 terms of the sequence:
1,9,17,25,33,41…
22. Find the smallest positive integer, , such that
by exactly 3 of the first 5 positive integers.
is divisible
23. Find the area of a regular octagon with a side length of 2
feet. Express your answer in simplest radical form.
2
24. If
find
24.
_
25. Yujian has a two digit number in the form AB. He knows
that
is a four digit number and can be written in the
form CDEF where
,
,
, and
. Find Yujian’s number.
25.
_
26. In Mr. Yodice’s Social Studies class there are a total of 45
students. Every single student in the class likes at least one
of three sports: football, baseball, or cricket. Exactly of the
class likes football, 28 people like
cricket, and 21 people like baseball.
If exactly 3 students like all three
sports, how many students like
exactly 2 sports?
26.
students
27. What is the value of
27.
_
28.
cents
29.
_
28. In the land of BLOINK there are two coins in the currency.
There is the ploink which is the 5¢ coin and there is the
kzoink which is the 11¢ coin. Find the largest number of
cents that cannot be made exact change for in the land of
BLOINK.
29. What is the largest possible product of a set of positive
integers, not necessarily distinct, that have a sum of 14?
30. Daniel loves bunnies. He has a huge rectangular backyard
which he wants to fill with bunnies. However the bunnies
he wants to buy will kill each other if put in the same
enclosed region. To make the regions he buys 11 long
straight fences which can extend as far as he wants (but
remember they are straight). If he wants each bunny to
have a separate region, and he uses all of the fences, what is
the greatest number of bunnies Daniel can buy?
Shown below is the greatest number of bunnies (7) Daniel
can buy with 3 fences. (I feel bad for the middle bunny).
30.
bunnies
MATHCOUNTS

®
KJ School Competition #2 
Target Round
Problems 1 and 2
Name _______________________________________________
Grade :__________
Teacher ____________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
This round of the competition consists of eight problems,
which will be presented in pairs. Work on one pair of
problems will be completed and answers will be collected
before the next pair is distributed. The time limit for each
pair of problems is six minutes. The first pair of problems
is on the other side of this sheet. When told to do so, turn
the page over and begin working. Record your final answers
in the designated space on the problem sheet. All answers
must be complete and legible. This round assumes the use
of calculators, and calculations may be done on scratch paper,
but no other aids are allowed.
In each written round of the competition, the required unit for the
answer is included in the answer blank. The plural form of the unit
is always used, even if the answer appears to require the singular
form of the unit. The unit provided in the answer blank is the only
form of the answer that will be accepted.
Total Correct
Scorer’s Initials
1.
2.
Haolin’s pet turtle has a square plot of land with an area of
10 square feet. If Haolin doubles the
length of each side of the square, how
many square feet will her turtle
have?
In the game of cricket, runs can be scored in 5
ways: singles, doubles, triples, 4’s, or 6’s. In
the Scholars Academy Cricket Match: Nabu
scored 7 6’s, 4 4’s, and 21 singles. If he scored
a total of 180 runs, find the greatest number
of triples Nabu could’ve hit.
1.
ft2
2.
triples
MATHCOUNTS

KJ School Competition #2 
Target Round
Problems 3 and 4
Name _______________________________________________
Grade :__________
Teacher ____________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
Total Correct
Scorer’s Initials
®
3.
A circle, a square, and a line are drawn in a plane. What is
the greatest number of distinct intersections the three
shapes could have?
3.
intersections
4.
m2
The diagram below shows 4 intersections (which isn’t the
most).
4.
I draw a circle such that its area, in square meters, is
exactly of its circumference. Find the area of the circle, in
square meters. Express your answer as a decimal to the
nearest thousandth.
MATHCOUNTS

KJ School Competition #2 
Target Round
Problems 5 and 6
Name _______________________________________________
Grade :__________
Teacher ____________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
Total Correct
Scorer’s Initials
®
5.
When 3 standard 6-sided die are rolled, what is the
probability that the sum of the upward faces is 6? Express
your answer as a common fraction. One instance is shown
below:
5.
_
6.
A cube is inscribed in a sphere of radius
inches. The
surface area of the cube, in square inches, can be expressed
in the form
where
and are positive
integers. Find
6.
_
MATHCOUNTS

KJ School Competition #2 
Target Round
Problems 7 and 8
Name _______________________________________________
Grade :__________
Teacher ____________________________
DO NOT BEGIN UNTIL YOU ARE
INSTRUCTED TO DO SO
Total Correct
Scorer’s Initials
®
7.
The numbers 1-9 will be put in the circles shown below
such that the sums of the triangle’s sides (4 circles) are
equal. What is the largest possible value of this sum?
7.
_
8.
Find the sum of all two digit numbers which satisfy the
following requirement: the product of the number’s digits
minus the number itself is equal to
.
8.
_
For example: the number 25 would become
MATHCOUNTS

®
KJ School Competition #2 
Team Round
Problems 1 – 10
School ________________________________________
Chapter________________________________________
Team Members______________________________________ , Captain
_______________________________________________
_______________________________________________
_______________________________________________
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED
TO DO SO.
This section of the competition consists of 10 problems which the team
has 20 minutes to complete. Team members may work together in any
way to solve the problems. Team members may talk during this section
of the competition. This round assumes the use calculators, and
calculations may also be done on scratch paper, but no other aids are
allowed. All answers must be complete, legible, and simplified to lowest
terms. The team captain must record the team’s official answers on his/her
own problem sheet, which is the only sheet that will be scored.
Total Correct
Scorer’s Initials
1.
What is the area of the triangle with vertices
and
?
1.
_
2.
Walter has the below configuration of rectangles and
squares. In total how many rectangles are there in Walter’s
configuration?
2.
rectangles
3.
Below is a
rectangular grid of unit squares with a
segment containing the midpoints of two squares. Find the
area of the shaded region. Express your answer as a
common fraction.
3.
_
4.
Bob has all the letters of BANANA in a bag (1 B, 3 A’s, and 2
N’s). He spills them on a table, decides to
pick up exactly 2 letters, and make a 2letter word with them. How many
distinct 2-letter words can Bob Bob
make?
4.
2-letter words
5.
_
BANANA
5.
What is the coefficient of the
is multiplied out?
term when
6.
The Cavaliers and the Celtics play a best of 3 series. Because
the Cavaliers are ranked higher than the Celtics, they are
given 1 more home game than the Celtics. The Cavs play at
home in the 1st game and the 3rd
game (if there is a 3rd game). The
Cavaliers win an away game with
the probability of and a home
6.
_
7.
cubes
8.
_
9.
_
game with the probability of .
Find the probability that the
Cavaliers will win the series.
Express your answer as a
common fraction.
7.
How many
cube to make a
cubes are needed to adjoin a
cube?
8.
Find the product of the greatest common divisor and least
common multiple of 1337 and 343.
9.
In Circle O,
and the
radius of the circle
is 6. Find the area
of the shaded
region. Express
your answer in
simplest radical
form in terms of .
10. ABCDEFGHIJKL is a regular dodecagon (12-sided polygon)
with side length
cm. If the vertices of the dodecagon
are labeled clockwise, find the area of triangle FGH, to the
nearest whole number
10.
cm2