Math Kangaroo is an international competition in solving mathematical problems, organized for students in grades 2 through 12. It is very popular in Europe and now is finding its way into the United States -- including Pittsburgh. No other scholarly mathematical competition in the world has such high participation: Over 5 million students from 41 countries took part in 2008. This year's Kangaroo will be held around the world on Thursday.

The goal is to solve 24 problems within 75 minutes. The tests are evaluated nationally and students with the 10 best scores in each grade level are recognized and win prizes. Calculators are not allowed in the Kangaroo, but the students may use basic geometric tools.

The problems you'll find in a Kangaroo are grade-appropriate, in that they require no special math skills beyond what's learned at school. But they are nothing like the run-of-the-mill problems the students normally tackle in class.

The challenge comes in several different flavors. Some problems can be solved using standard arithmetic procedures, but such a solution is a long and tedious undertaking that would make it impossible to finish the test in time. On the other hand, with a little imagination and logical deduction, one can discover a hidden law or a rule governing the results that can be applied to solve the problem in a much shorter time.

Other problems require the student to use logical deduction to infer the result from initial information that, at first sight, looks incomplete. Yet other problems are based on challenging ingrained stereotypes of human thinking, requiring the student to think "outside the box."

The students enjoy solving these problems not because they are mathematical, but because they make them think hard. To them, problem-solving is not a chore, but a game to play -- when much too often they are required only to learn and repeat schemes and procedures without thinking.

And why use mathematics? Because mathematics is not just a way to work with numbers. It is a tool that has been developed by humans to allow precise formulation of ideas and questions, and an unbiased evaluation of the correctness of answers to those questions.

In mathematical competition, every answer is either true or false. Nothing is left in doubt or subject to interpretation.

The real challenge is having the courage to answer difficult questions and have your answers evaluated. It gives you a chance to see how smart you really are -- and provides a bit of pride for those who succeed and a dose of humility for those who do not.

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The Kangaroo in Pittsburgh, America and the world

Pittsburgh's first Kangaroo took place in 2007 with eight participants. The next year their number swelled to 131. This year's competition will take place on Thursday at 6 p.m in the University of Pittsburgh's Cathedral of Learning.

Participants had to preregister by January. But for more information, and to get ready for next year, go to www.pitt.edu/~hajlasz/Kangur/kangaroo.html

The first U.S. Kangaroo was held in 1998 in Chicago. Last year, some 2,100 students in about 20 states participated (see www.mathkangaroo.org for more information).

Russia has the highest participation with more than 1.63 million students last year, followed by Germany at 768,000 and France at 340,000. Puerto Rico had more than 6,500. The United States was outshone by Mexico, which enlisted 8,600 students, but we crushed the Canadians, who could muster only 778.

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Why is it called a 'Kangaroo'?

"In the early 1980s, Peter O'Holloran, a math teacher in Sydney, Australia, invented a new kind of game in Australian schools: a multiple choice questionnaire, corrected by computer, which meant that thousands of pupils could participate at the same time. It was a tremendous success for the Australian Mathematical National Contest.

In 1991, two French teachers (Andre Deledicq and Jean Pierre Boudine) decided to start the competition in France under the name 'Kangaroo' to pay tribute to their Australian friends."

-- From the Web site of Kangourou Sans Frontieres, the French organization that promotes the event worldwide (www.math-ksf.org)

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Here are some problems from each level of the competition, selected from past Kangaroos. Click here to download the problems on a page in .pdf format. Within each level, the problems become progressively more difficult:

Problem 1

Only one digit from 1 to 9 is repeated three times in this drawing. The rest of the digits are repeated twice. Which digit is repeated three times?

A) 9
B) 8
C) 3
D) 4
E) 7

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Problem 2

How many more square tiles do we need to put on the kitchen floor to cover all of it? (See the picture.)

A) 12
B) 10
C) 9
D) 6
E) 4

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Problem 3

A train has four cars in four colors: red, green, white and yellow. The green car is not the first nor the last. The yellow car is not next to the white car nor next to the red car. The first car is white. What is the order of the cars in that train?

A) white, green, red, yellow
B) white, yellow, green, red
C) green, yellow, red, white
D) red, white, green, yellow
E) white, red, green, yellow

 

Problem 1

After we simplify 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2, what will be the result?

A) 0
B) 2
C) 4
D) 12
E) 20

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Problem 2

During a Kangaroo camp, Adam solved five problems every day, and Brad solved two problems daily. After how many days did Brad solve as many problems as Adam solved in 4 days?

A) After 5 days
B) After 7 days
C) After 8 days
D) After 10 days
E) After 20 days

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Problem 3

There were 9 pieces of paper. Some of them were cut into three pieces. As a result, there are 15 pieces of paper now. How many pieces of paper were cut?

A) 2
B) 3
C) 4
D) 5
E) 6

 

Problem 1

Which of the following numbers is not a factor of 2004?

A) 3
B) 4
C) 6
D) 8
E) 12

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Problem 2

How many shortest distances along the edges of the cube are there that connect vertex A with the opposite vertex B?

A) 4
B) 6
C) 3
D) 12
E) 16

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Problem 3

Symbols P, Q, R, S indicate the total weight of the figures drawn above them.

It is known that any two figures of the same shape have the same weight. If P < Q < R, then:

A) P < S < Q
B) Q < S < R
C) S < P
D) R < S
E) R = S

 

Problem 1

A square built of 16 small squares is cut with a line. What is the greatest number of the little squares that the line can go through?

A) 3
B) 4
C) 6
D) 7
E) 8

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Problem 2

Points P and Q are the centers of two outside tangent circles (see the picture.) The line going through points P and Q intersects these circles at points A and B. If the area of rectangle ABCD is 15 then what is the area of triangle PQT?

A) 4
B) 15/4
C) π/2
D) 5
E) 5 ÷ 2

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Problem 3

For each triplet of numbers (a, b, c) one can create another triplet (b + c, c + a, a + b). This is called an operation. After performing 2004 such operations starting with numbers (1, 3, 5), one obtains the triplet (x, y, z). What is the difference x - y equal to?

A) -2
B) 2
C) 4008
D) 2004
E) (-2)2004

 

Problem 1

Which of the following numbers is odd for every integer n?

A) 2003n
B) n² + 2003 [NOTE: n² is "n to the power of 2"]
C) n³ [NOTE: n³ is "n to the power of 3"]
D) n + 2004
E) 2n² + 2003 [NOTE: 2n² is "2n to the power of 2"]

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Problem 2

Using 3 bricks each consisting of 4 little cubes, a rectangular parallelepiped has been built (see picture). The crosshatched brick is completely visible; both others -- partly visible. Which brick is the dark one?

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Problem 3

The children A, B, C and D made the following assertions:

A: "B, C and D are girls"
B: "A, C and D are boys"
C: "A and B are lying"
D: "A, B and C are telling the truth"

How many of the children were telling the truth?

A) 0
B) 1
C) 2
D) 3
E) It can't be determined

 

Problem 1

The average number of students accepted by a school in the four years 1999-2002 was 325 students per year. The average number of students accepted by the school in the five years 1999-2003 is 20 percent higher. How many students did this school accept in 2003?

A) 650
B) 600
C) 455
D) 390
E) 345

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Problem 2

Let ABC be a triangle with area 30. Let D be any point in its interior and let e, f and g denote the distances from D to the sides of the triangle. What is the value of the expression 5e + 12f + 13g?

A) 120
B) 90
C) 60
D) 30
E) It is not possible to determine the value without knowing the exact location of D.

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Problem 3

Let A and B be positive integers, such that A > B > 1, and A, B, A-B, A+B are all prime. Then S = A + B + (A-B) + (A+B)

A) is even
B) is a multiple of 3
C) is a multiple of 5
D) is a multiple of 7
E) is prime

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The Answers

Level 2
1) E 2) C 3) E

Level 3-4
1) C 2) D 3) B

Level 5-6
1) D 2) B 3) A

Level 7-8
1) D 2) B 3) A

Level 9-10
1) E 2) A 3) B

Level 11-12
1) A 2) C 3) E

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Correction/Clarification: (Published Mar. 22, 2009) In this Next Page article "Math Kangaroo," as originally published Mar. 15, 2009, Problem 1 in Level 9-10 was presented incorrectly. Answer B should have been n² + 2003 (not n2). Answer C: n³ (not n3). Answer E: 2n² + 2003 (not 2n2).