Math Stars H 2 7 D I 3 8 E J 4 9 TotalCorrect Scorer’sInitials CSSMA Major Sponsors...

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Saturday, June 4th, 2016F National ChampionshipF

Team Relays RoundProblem Set A: Algebra

School/Team Code Grade(s)

Team Members

Team Captain

DO NOT BEGIN UNTIL YOU ARE INSTRUCTED TO DO SO.

Number of Problems: 5 in this problem setTime Allotted: 45 minutes for all five problem sets combined

Scientific calculators are permitted, but books or other aids are not permittedAnswer in exact form (i.e. integer, common fraction...etc.) and round only when asked to do so.No units need to be provided after your answers. Please record only final answers in the blanks inthe left-hand column of the competition paper. If your team completes the problems before time iscalled, use the remaining time to check your answers.

Form Code

A F 0 5B G 1 6C H 2 7D I 3 8E J 4 9

Total Correct Scorer’s InitialsCSSMA Major

SponsorsUniversity of Toronto

UBC Math ClubCanadian Mathematical Society

Expii.inc.Various PAC committees

Copyright Canadian Secondary School Mathematics Association (CSSMA). 2015-2016. All rights reserved.

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Problem Set A: Algebra1. ________________ 1. Steven planted the same number of trees every day for three days. In

total, he planted 150 trees. How many trees did he plant per day?

2. ________________ 2. Solve for x:x+ 3a− 6 = 3x− 288

Note: a is the answer to the previous problem.

3. ________________ 3. Albert has 3√b + 8 apples and Carl has 3

√b + 4 apples, where b is the

answer to the previous problem. How many apples must Albert give toCarl so that Carl has three times the number of apples as Albert?

4. ________________ 4. I have c2 − 10 red, green, and blue balls in total, where c is the answerto the previous problem. There are twice as many green balls as redballs and three times as many blue balls as red balls. How many moreblue balls are there than red balls?

5. ________________ 5. 5d+10 birds eat 6d− 8worms. Big birds eat√d− 9worms each, and√

d− 9 small birds eat 1 worm together, where d is the answer to theprevious problem. How many small birds are there?


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Team Relays RoundProblem Set B: Geometry


Team Members

Team Captain




Form Code

A F 0 5B G 1 6C H 2 7D I 3 8E J 4 9






Problem Set B: Geometry1. _______________° 1. What’s the sum of the internals angles of a square, in degrees?

2. ________________ 2. On line segment AB, point C is chosen so that AC = 2BC. On linesegment AC, pointD is chosen so that AD = CD. If AB = a, wherea is the answer to the previous problem, then what’s the length of CD?

3. ________________ 3. The surface area of a cube is 2b + 54, where b is the answer to theprevious problem. What’s the volume of this cube?

4. ________________ 4. The side lengths of a triangle are three consecutive positive even inte-gers. The perimeter of the only right-angled triangle that satisfies thiscriteria is m. What’s the remainder when c is divided by m, if c is theanswer to the previous problem?

5. ________________ 5. A right square pyramid has a square base of side length d − 5, whered is the answer to the previous problem. All faces of this pyramidare regular polygons. The total surface area of this pyramid can beexpressed in the form x+ y

√z where x, y, z are positive integers and z

is not divisible by any perfect square greater than 1. What’s the valueof x+ y + z?


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Team Relays RoundProblem Set C: Number Theory


Team Members

Team Captain




Form Code

A F 0 5B G 1 6C H 2 7D I 3 8E J 4 9






Problem Set C: Number Theory1. ________________ 1. Jeffery has 50 books. He stacks them in piles of 6 until he can’t do so

any more. How many books are left over?

2. ________________ 2. A pattern of shapes is written out:

4 © � ♦ 4 © � ♦ 4 © � ♦ · · ·

Of the first 20 · a shapes, how many are quadrilaterals? Quadrilater-als are polygons with four sides, and a is the answer to the previousproblem.

3. ________________ 3. How many prime numbers are there which are less than b, where b isthe answer to the previous problem? Prime numbers have only twopositive integer divisors.

4. ________________ 4. A sequence of numbers called the Fibonacci sequence starts with 1,1, 2, 3, 5, 8... Starting from the third term, every term is equal to thesum of the previous two terms. For example, 1+1=2, 1+2=3, 2+3=5,3+5=8...etc.

What’s the smallest Fibonacci number that has a units digit ofc− 1, where c is the answer to the previous problem?

5. ________________ 5. How many even positive divisors does (d − 17) ÷ 9 have, where d isthe answer to the previous problem?


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Team Relays RoundProblem Set D: Combinatorics & Probability


Team Members

Team Captain




Form Code

A F 0 5B G 1 6C H 2 7D I 3 8E J 4 9






Problem Set D: Combinatorics & Probability1. ________________ 1. A fair 6-sided dice is rolled. What’s the probability that the top face is

even? Express your answer as a common fraction. Use "/" to denotethe bar between the numerator and the denominator.

2. ________________ 2. If a is the denominator of the answer to the previous problem, howmany a digit positive integers are there?

3. ________________ 3. A certain Regional contest has 100 ·b participants, where b is the answerto the previous problem. (b÷ 3)% of the Regional contest participantsqualified for a Provincial contest, and 20% of the Provincial contestparticipants qualified for a National contest. What is the probabilitythat a randomly chosen Regional contest participant will have qualifiedfor the National contest? Express your answer as a common fraction.Use "/" to denote the bar between the numerator and the denominator.

4. ________________ 4. Jesse writes down the digits of the natural numbers in increasing order,like so: 1, 2, 3, 4, 5,. . .. What’s the cth digit that he’ll write, if c is thedenominator of the answer to the previous problem? For example, the11th digit that he’ll write is the "0" in the number 10.

5. ________________ 5. A quiz has d+1multiple-choice questions with d+1 answer options foreach question (of which exactly one is correct), where d is the answerto the previous problem. Meghan takes the quiz and guesses eachquestion. What is the probability that she will get at least one questionright (i.e. she will not get all of the problems wrong)? Express youranswer as a common fraction. Use "/" to denote the bar between thenumerator and the denominator.


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Team Relays RoundProblem Set E: Logic


Team Members

Team Captain




Form Code

A F 0 5B G 1 6C H 2 7D I 3 8E J 4 9






Problem Set E: Logic1. ________________ 1. A group of three friends are talking about their jobs. One of them is

a manager, one of them is an accountant, and one of them is a truckdriver. It is known that:

• Catherine is older than the accountant• Alyssa’s age is different from the truck driver’s age• The truck driver is younger than Beryl

Which of the three is the manager?

2. ________________ 2. Alyssa, Beryl, and Catherine from the previous problem now have aconversation about a donation that one of them made:

• Alyssa says: "the manager donated"• Beryl says: "I didn’t donate"• Catherine says: "I didn’t donate either"

If only one of the three statements above is true, who donated? Givethe name of the person.

3. ________________ 3. Alyssa, Beryl, andCatherine from the previous problems are now joinedby Daniel and Erin. They compete in a race. 5 of their friends predictthe results of the race (there are no ties):

• Friend 1 says: "The accountant will be 2nd; the manager will be3rd"

• Friend 2 says: "The person who donated will be 3rd; Daniel will be5th"

• Friend 3 says: "Daniel will be 1st; the truck driver will be 2nd"• Friend 4 says: "The accountant will be 2nd; Erin will be 4th"• Friend 5 says: "The manager will be 1st; Erin will be 4th"

It turns out that each friend only got one of their two predictions correct.Which person came 1st? Give the name of the person.

Problems 4 and 5 are on the next page =⇒


4. ________________ 4. Alyssa, Beryl, Catherine, Daniel, and Erin from the previous problemare now joined by five other individuals: Felicia, George, Hazel, Ian,and Jason. They pair up in teams of 2 for a trivia contest. The contesthas 4 rounds, and in each round, only one person from each pair canparticipate. It is known that:

• Round 1 participants: the individuals who got 1st, 3rd, and 4th inthe race, Felicia, and Hazel

• Round 2 participants: the individuals who got 3rd and 5th in therace, Felicia, Hazel, and Jason

• Round 3 participants: the individuals who got 2nd, 3rd, 4th, and 5thin the race, and Ian

• Round 4 participants: the individuals who got 1st, 2nd, and 3rd inthe race, Felicia, and Jason

It turns out that George was sick so he didn’t participate in any of the4 rounds. Who was Erin’s partner? Give the name of the person.

5. ________________ 5. It’s 10 years later, and all 10 individuals from the previous problemhave changed their jobs. All of the 10 individuals are now at a party.Of them, Alyssa, Beryl, Catherine, Daniel, Erin, and Felicia are ofdifferent nationalities. It is known that:

• There’s exactly one American, one English, one French, one Ger-man, one Russian, and one Canadian in the group

• Daniel’s partner in the trivia contest and the American are doctors• Jason’s partner in the trivia contest and the Russian are teachers• Hazel’s partner in the trivia contest and the German are architects• George and Ian’s partners in the trivia contest are athletes• The German does not like sports• The French is older than Daniel’s partner in the trivia contest• Hazel’s partner in the trivia contest is younger than the Canadian• George’s partner in the trivia contest and the American want totravel out of their home countries to Britain for a vacation

• Hazel’s partner in the trivia contest and the French want to travelto Italy for work next year

Who must be French? Give the name of the person.


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