Math Stars
®
2016
F National Championships F
Sample Sprint Round
Problems 1 – 30
HONOUR PLEDGE
I pledge to uphold the highest principles of honesty and integrity as a Math Stars competitor. I will neither give nor
accept unauthorized assistance of any kind. I will not copy another’s work and submit it as my own. I understand
that any competitor found to be in violation of this honour pledge is subject to disqualification.
Signature
Date
Printed Full Name
School/Team Code
Province/Territory
DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.
This section of the competition consists of 30 problems. You will have 40 minutes to complete all the problems. You
are not allowed to use calculators, books or 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 left-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.
Form Code
A
F
B
G
C
H
D
I
E
J
CSSMA Sponsors
0
1
2
3
4
5
6
7
8
9
Total Correct
Scorer’s Initials
UBC Math Club
Canadian Mathematical Society
Expii, Inc
Other sponsors TBA
The CSSMA would like to express its gratitude to the MATHCOUNTS, Inc. for the wording of the Honour Pledge, competition instructions, and the
format for the title page for this competition booklet, as shown above, as well as the Forms of Answers document on the rear of this test booklet.
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
THIS PAGE IS INTENTIONALLY LEFT BLANK
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
1. ____________________
1. If 4 workers can make 12 clarinets in 3 hours, how many clarinets
can 1 worker make in 20 hours?
2. ____________________ 2. How many revolutions does it take for a wheel with radius
to travel 800m?
16
π cm
3. Eric is mixing fruit juices. Cup 1 has 12ml of orange and apple
juice, in a ratio of 1:3, respectively. Cup 2 has 24ml of orange
and apple juice, in a ratio of 1:2, respectively. If Eric pours 6ml
3. ____________________
of juice from cup 2 to cup 1, what will the new ratio of orange to
apple juice be in cup 1? Express your answer in the format a : b,
and NOT ab .
4. Anne’s plane leaves Hong Kong at 6:00 PM (Hong Kong time)
and arrives in Vancouver at 10:00 PM (Vancouver time). When it
4. _______________ hours
is midnight in Vancouver, it is 3:00 PM in Hong Kong. How long
was Anne’s flight, in hours?
5. ____________________
5. What is the intersection point of the lines y−2x = 5 and 3y+4x =
10?
6. ____________________
6. What is the area of a rhombus with perimeter 80 and two interior
angles of 60°?
7. ____________________ 7. How many integers N are there such that
49
5+2N
is an integer?
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
8. ____________________
8. The digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 must all be arranged into
three 3 digit numbers and added (for example, 123 + 456 + 789
= 1368). What is the smallest such possible sum?
9. A circle of radius 8 and a circle of radius 4 are tangent to each
other. A tangent from the center of the circle of radius 8 to the
9. ____________________
circle of radius 4 is drawn. What is the length from the point of
tangency to the center of the circle of radius 8?
10. A ball with radius 3 is dropped into a cylinder of radius 4 and
height 20. The initial height of the water is 10. What is the height
10. ____________________
of the water after the ball is added, assuming the ball fully sinks?
11. During the French Revolution, the French tried to replace the 24
hour clock with a decimal time system. In decimal time, a day
is 10 hours, an hour is 100 minutes and a minute is 100 seconds.
11. ____________________
When it is 6:43:22 AM, what time is it in decimal time? Round
your answer down to the nearest second.
12. How many zeroes are in the product of:
12. ____________________
. . 999}?
111
. . 111} × |999 .{z
| .{z
21 1’s
21 9’s
13. I take two standard dice and add two dots to each face, and then
roll the pair of dice. What is the probability that when I multiply
13. ____________________
the two numbers that show up on the dice (the number on a face
is equal to the number of dots on that face), the product is 24?
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
14. ________________ m/s
14. Norman swims at 6m/s, bikes at 36m/s, and runs at 12m/s. He
does each of the activities for the same distance. What is his
average velocity?
15. Two acid solutions are mixed together. Solution A is 20% acid
by volume and Solution B is 60% acid by volume. How much of
15. _________________ ml
Solution A, in ml, is needed to mix with Solution B to make a 500
ml mixture that is 50% acid by volume?
16. 9 balls numbered with the integers 1 through 9 inclusive (with
1 ball per number) are placed in a raffle spinner. What is the
16. ____________________
probability that when 3 balls are drawn without replacement, the
first ball is odd, the second even, and the third odd?
17. A man dies between the year 1320 and 1350. The age of his
1 th
17. _______________ years
death is exactly 25
of his year of birth. What age (in years) does
he turn in the year 1313?
18. Alphonse, Beryl, Carl, Daniel, and Elaine are each on a vertex
of a regular pentagon, with one person on each vertex. Every
minute, each of them randomly chooses an adjacent vertex to go
18. ____________________
to with equal probability. What’s the probability that after four
minutes, they are all back on their original vertex? Express your
answer as a common fraction.
19. How many of the permutations that can be made using one or
19. ____________________
more letters in "VANCOUVER" also can be made using the letters in "CANADA"?
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
20. How many quadrilaterals (which are not 4 points on a line or a
triangle) can you make from picking 4 lattice points on a 2 × 2
grid, composed of four 1 × 1 small squares? Note: lattice points
20. ____________________
are where two or more line segments meet, and there are 9 points
to choose as possible vertices for the quadrilateral.
21. ____________________
21. How many pairs of three digit perfect squares have the same tens
digit and units digit?
22. ____________________
22. How many license plates are possible if a license plate consists
of 3 letters and 3 digits (each from 0 to 9 inclusive), in any order?
23. In the following figure, A0 is the midpoint of AB, B 0 is
the midpoint of BC, C 0 is the midpoint of CD, and so on.
ABCDEF GH is a regular octagon. What is the ratio of the area
of A0 B 0 C 0 D0 E 0 F 0 G0 H 0 to the area of ABCDEF GH?
23. ____________________
24. ____________________
24. The six digit positive integer 11ab11 is divisible by 17 and 19.
What is the value of the two digit integer ab?
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
25. ____________________
25. Alphonse is typing the numbers from 1−10000. However, since
the key for 2 is broken, he skips all the numbers containing the
digit 2. What is the 500th number he will type?
26. A cube of edge length 1 is dangling from a string attached to one
of its vertices. The bottom-most vertex is just touching a horizontal table. What height about the table are the vertices attached to
26. ____________________
the top-most vertex of the cube (which is the vertex attached to
the string)?
27. In square ABCD of side length 1, point M is the midpoint of
CD. Let N be the midpoint of AM , O be the midpoint of BN ,
27. ____________________
P be the midpoint of CO, and Q be the intersection point of AM
and DP . What is the area of quadrilateral N OP Q?
28. ____________________ 28. Determine the greatest common divisor of 106 − 1 and 610 − 1.
29. Find the largest possible value of x + y if the following system
satisfies:
(
29. ____________________
x + y1 = 2
y + x1 = 4
30. 1g of gold is worth $10, 1g of silver is worth $5, and 1g of bronze
is worth $1. $220 is used to purchase 100g of gold, silver, and
30. ____________________
bronze. A positive integer number of grams of each metal is purchased. How many possible combinations of purchase are there?
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.
Forms of Answers
The following list explains acceptable forms for answers. Coaches should ensure that Math Stars competitors are familiar with
these rules prior to participating at any level of competition. Judges will score competition answers in compliance with these
rules for forms of answers.
All answers must be expressed in simplest form. A “common fraction” is to be considered a fraction in the form ± ab ,
where a and b are natural numbers and gcd(a, b) = 1. In some cases the term “common fraction” is to be considered a fraction
A
in the form B
, where A and B are algebraic expressions and A and B do not share a common factor. A simplified “mixed
number” (“mixed numeral,” “mixed fraction”) is to be considered a fraction in the form ±N ab , where N , a and b are natural
numbers, a < b and gcd(a, b) = 1. Examples:
Problem: Express 8 divided by 12 as a common fraction.
Answer: 23
Unacceptable: 64
1
Problem: Express 12 divided by 8 as a common fraction.
Answer: 23
Unacceptable: 12
8 , 12
Problem: Express the sum of the lengths of the radius and the circumference of a circle with a diameter of 41 as a common
fraction in terms of π.
Answer: 1+2π
8
8 5
Problem: Express 20 divided by 12 as a mixed number.
Answer: 1 32
Unacceptable: 1 12
,3
Ratios should be expressed as simplified common fractions unless otherwise specified. Examples:
Simplified, Acceptable Forms: 72 , π3 , 4−π
6
1
Unacceptable: 3 21 , 34 , 3.5, 2 : 1
Radicals must be simplified. A simplified radical must satisfy: 1) no radicands have a factor which possesses the root indicated by the index; 2) no radicands contain fractions; and 3) no radicals appear in the denominator of a fraction. Numbers
with fractional exponents
are
√
√ not in radical form. Examples:
√
√
Problem: Evaluate 15 × 5.
Answer: 5 3
Unacceptable: 75
Answers to problems asking for a response in the form of a dollar amount or an unspecified monetary unit (e.g., “How
many dollars...,” “How much will it cost...,” “What is the amount of interest...”) should be expressed in the form ($) a.bc,
where a is an integer and b and c are digits. The only exceptions to this rule are when a is zero, in which case it may be
omitted, or when b and c are both zero, in which case they may both be omitted. Answers in the form ($)a.bc should be rounded
to the nearest cent unless otherwise specified. Examples:
Acceptable: 2.35, 0.38, .38, 5.00, 5
Unacceptable: 4.9, 8.0
√
Do not make approximations for numbers (e.g., π, 32 , 5) in the data given or in solutions unless the problem says to
do so.
Do not perform any intermediate rounding (other than the “rounding” a calculator does) when calculating solutions. All
rounding should be done at the end of the computation process.
Scientific notation should be expressed in the form a × 10n where a is a decimal, 1 < |a| < 10, and n is an integer. If
a can be expressed as an integer, either a × 10n or a.0 × 10n will be accepted. Examples:
Problem: Write 6895 in scientific notation.
Answer: 6.895 × 103
Problem: Write 40,000 in scientific notation.
Answer: 4 × 104 or 4.0 × 104
An answer expressed to a greater or lesser degree of accuracy than called for in the problem will not be accepted.
Whole number answers should be expressed in their whole number form.
Thus, 25.0 will not be accepted for 25, and 25 will not be accepted for 25.0.
Units of measurement are not required in answers, but they must be correct if given. When a problem asks for an
answer expressed in a specific unit of measure or when a unit of measure is provided in the answer blank, equivalent answers
expressed in other units are not acceptable. For example, if a problem asks for the amount in kilograms and 2 kg is the correct
answer, 2000 g will not be accepted. If a problem asks for the number of cents and 25 cents is the correct answer, $0.25 will
not be accepted.
The plural form of the units will always be provided in the answer blank, even if the answer appears to require the
singular form of the units.
Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.