Canadian Math Challengers Society Question Archive

980 Pages·2015·9.12 MB·English

Canadian Math Challengers Society
Exam Archive
2005 to 2019
2005 Regional and Provincial Questions
2006 Regional and Provincial Questions
2006 Regional and Provincial Answers
2007 Regional and Provincial Questions
2007 Regional and Provincial Answers
2008 Regional and Provincial Questions
2008 Regional and Provincial Answers
2009 Regional and Provincial Questions
2009 Regional and Provincial Answers
2010 Regional and Provincial Questions
2010 Regional and Provincial Answers
2011 Regional and Provincial Questions
2011 Regional and Provincial Answers
2012 Regional and Provincial Questions
2012 Regional and Provincial Answers
2013 Regional and Provincial Questions
2013 Regional and Provincial Answers
2014 Regional and Provincial Questions face off missing
2014 Regional and Provincial Answers face off missing
2015 Regional and Provincial Questions
2015 Regional and Provincial Answers
2015 Regional and Provincial Questions
2016 Regional and Provincial Answers
2017 Regional and Provincial Questions
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Canadian Math Challengers Society
Exam Archive

2018 Regional Questions
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Page 1: Problem Solving
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1. A class of 30 students took a test, and the class average was 70. The 1.
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five students who failed had marks of 20, 25, 25, 30, and 40. What
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was the averagemark among the students who didn’tfail? Give the
n answercorrect to 1decimalplace.

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t 2. BetahastwiceasmanypenniesasAlpha. Gammahasthreetimesas 2.
h many pennies as Beta. Between the three of them, they have fewer
than80pennies. Whatis thelargest possiblenumberof penniesthat

theycould havebetweenthem?
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l 3. Xaviera jogs at a steady rate of 5 minutes per kilometre. Yolande 3.
e jogs at 7 minutes per kilometre. They start at the same time on an
out and back run that consists of 15 km due east followed by 15 km
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duewest. HowmanykilometresfromthefinishlineisXavierawhen
they pass next to each other? Give the answer correct to 1 decimal
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place.
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4. It so happensthat 7700625is a perfect square, that is, the square of a 4.
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whole number, and that 7706176 is the next perfect square. What is
i thefirst perfect square greater than7706176?
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Page 2: Combinatorics
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5. In a game, you toss two standard dice, a silver one and a gold one. 5.
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Your score is the number showing on the silver one, plus twice the
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number showing on the gold one. What is the probability that your
score is 8? Express your answeras a fraction inlowest terms.
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h 6. There are 5 people in a family, all of different heights. We want to 6.
linethem up for a picture, with the tallest person inthe middle,and
so that as we go from left to right, the heights of the people increase
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andthen decrease. Howmany ways are there to do this?
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7. Let A, B, and C be the measures, in degrees, of the angles of a tri- 7.
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angle, where A B C. How many possibilities are there for the
ordered triple (A,B,C), given that each of A, B, and C is a multiple
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of 15?
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8. How many4-digitnumbersarethere whichobey allthree of thefol- 8.
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lowing rules: (i) no digit other than 1, 2, 3, or 4 is to be used; (ii) a
digit may occur more than once; (iii) as you read the number from
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leftto right, digits neverdecrease(so 1134is OK, but 3314is not)?
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Page 3: Geometry
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9. A regular polygon has 11 sides. How many diagonals does it have? 9.
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A diagonal is a line segment that joins two corners but is not a side.
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Onediagonalis shown inthe picture.
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10. Lines that look parallel in the picture are parallel. Express b, the 10.
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lengthof the“middle”linesegment, as acommon fraction.

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e 11. The figure below is a regular hexagon with area 1. Express the area 11.
of the shadedregion as acommon fraction.
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12. Each side of triangle ABC has length 2. Three semicircles are con- 12.
S structed outside ABC withthe three sides AB, BC, CA as diameters.
A thread PQRSTUP is tied tightly around the resulting “three-leaf
o clover” shapeso that PQ, RS, TU are arcs of the semicircles withdi-
ameters AB, BC,CA respectivelyand QR, ST,UP arelinesegments.
c Find the area enclosed by the thread. Give your answer in terms of
(cid:25).
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Page 1
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1. What is the largest number that is less than 100 and has exactly 3 1.
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wholenumberfactors?
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2. Triangle ABC has a right angle at A. The two legs AB and AC have 2.
lengths 9 centimetres and 40 centimetres. Whatis the length, in cen-
a timetres, of the hypotenuse BC?
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1 3 3 5
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2x 2 4x 4
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4. If a computer is worth a certain amount, it is worth 40% less a year 4.
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l later. Alphonse’s computer is now worth $2000. How many dollars
e willit be worth 3 years from now?
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e 5. Suppose that a and b are real numbers such that ab = 5. What is the 5.
valueof a3b −6?
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6. The cost of sending a parcel is $4.00 for the first kilogram, and $0.60 6.
for each additional kg. A certain parcel weighs a whole number of
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kg, and costs $40.00to send. How many kg doesthe parcelweigh?
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t 7. Arectangularfieldis50%longerthanitiswide. Theperimeterofthe 7.
y fieldis 300metres. Whatis the areaof the field, in square metres?
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Page 2
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8. Alphonse sold a house to Beth for $300,000. She sold it immedi- 8.
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ately to Gamal at a 10% profit. Then Gamal sold the house back
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to Alphonse at a 10% loss. What was Alphonse’s overall profit (in
dollars)?
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9. Astringoflength120centimetresiscutintothreepartswhoselengths 9.
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are proportional to 4, 5, and 6. What is the length, in centimetres, of
thelongest part?
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10. To get some money, Tom decided to sell his CDs. After he had sold 10.
six-sevenths of his collection for $8.00 per CD, he had three CDs left
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that hecouldn’t sell. How much moneydid Tom get (indollars)?
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11. The ratio of x to y is , and the ratio of x to z is . What is the ratio 11.
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of y to z? Express your answeras a common fraction.
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r 12. When an integer n is divided by 12, the remainder is 7. What is the 12.
s remainderwhen7n is dividedby 12?

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13. The two legs of a right-angled triangle have length 20 and 100. To 13.
thenearestinteger, what is the lengthof thehypotenuse?
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14. If a car travels at 70 kilometres per hour, how many metres does it 14.
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travelin 18 seconds?
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Page 3
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15. There are 12 tickets (numbered 1 to 12) in a hat. Alfonso takes two 15.
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tickets, chosen at random. What is the probability that the sum of
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the numbers on Alfonso’s two tickets is odd? Express your answer
as a commonfraction.
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16. Thenumberof cubic millimetres in a cubic kilometre is 10n. What is 16.
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n?
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17. Line segments PA, AB, BC, CD, DE, and EF have length 1, and an- 17.
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gles PAB, PBC, PCD, PDE, PEF are right angles. Find the length of
PF.
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E 1
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18. The sum of two positive whole numbers is 144. If the larger of the 18.
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two numbers is divided by the smaller, the quotient is 3 and the re-
mainderis 12. Whatis the smallerof thetwo numbers?
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19. Suppose that you play the following game: you toss a fair nickel, 19.
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dime, and quarter at the same time. If you get at least one “head,”
stop(gameover). Ifyou don’t,you toss thecoinsagain. Ifyou get at
S leastone head,stop. Otherwise, go on ....
Whenyou toss for the lasttime,what is the probability that allthree
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coins show heads? Express your answeras a common fraction.
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20. If we start adding theconsecutive positive integers likethis, 1+2+ 20.
t 3+4+5(cid:1)(cid:1)(cid:1), andwestop addingwhenthenextnumberwould put
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our sum over1000, what sum do we get?
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Page 4
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21. There were three candidates for mayor of Mathville, Alpha, Beta, 21.
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and Gamma. Gamma came in last with 5000 votes. Alpha was 1000
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votes ahead of Beta, who was 1500 votes ahead of Gamma. How
many peoplevoted?
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22. Let N = 222. What is the second digit from the left in the decimal 22.
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expansion of N? (If instead we had N = 38, then the answer would
t be 5,since 38 = 6561.)
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23. Alphonse lost all his marbles. Some were blue, some were white, 23.
h and the rest were red. All but 99 were blue, all but 85 were white,
andallbut 70were red. Howmany marblesdidAlphonselose?
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24. How many three-digit numbers have exactly one 9 in their decimal 24.
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expansion?
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25. Bethisone-fifthofthewaythroughhercross-country race. Aftershe 25.
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runs a further three-quarters of a kilometre, she will be one-quarter
ofthewaythroughtherace. Overhowmanykilometresisthewhole

S race?
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26. A gambler started off with 1 dollar. She placed a series of 1 dollar 26.
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e bets, winning a dollar or losing a dollar each time. After a total of
9 bets, the gambler was broke. In how many orders could this have
t happened? Ifyou haveno money you can’tbet.
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1. The product of three consecutive positive integers is equal to 4080. 1.
d Whatis the sum of thethree integers?
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2. In the picture below, the two smaller circles are equal in size, go 2.
a through the center of the larger circle, and are tangent to each other
and to the larger circle. If the area of each smaller circle is 17 square
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units,what is the area, in square units,of theshadedregion?
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l 3. How many integers n are there such that 1 (cid:20) n (cid:20) 64 and nn is a 3.
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perfectsquare?
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4. For any wholenumber n, the number n! is definedby 4.

S n! = (1)(2)(3)(4)(cid:1)(cid:1)(cid:1)(n−1)(n).
o For example4! = (1)(2)(3)(4) = 24.
What is the the remainder when 2!+3!+4!+(cid:1)(cid:1)(cid:1)+89!+90! is di-
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videdby 90?
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