加拿大国际袋鼠数学竞赛试题-2013年

Grade 1-2
International Contest-Game MATH KANGAROO
Part A: Each correct answer is worth 3 points. 1. Which digits are missing?
Year 2013
(A) 3 and 5 (B) 4 and 8
(C) 2 and 0
(D) 6 and 9
(E) 7 and 1
2. There are twelve books on a shelf and four children in a room. How
many books will be left on the shelf if each child takes one book?
(A) 12
(B) 8
(C) 4
(D) 2
(E) 0
3. Which of the dresses has less than seven dots, but more than five dots?
(A)
(B)
(C)
(D)
(E)

Grade 1-2
Year 2013
4. A lot of babies were born in the zoo last year: two baby lions, three baby dolphins and four baby eagles.
How many legs do all these babies have altogether?
(A) 20
(B) 18
(C) 16
(D) 14
(E) 12
5. Several students want to plant 20 tulips in the school garden. It takes ten minutes for them to plant five
tulips. They started at 9:00 in the morning. At what time will they finish planting all 20 tulips?
(A) At 9:10
(B) At 9:20 (C) At 9:40
(D) At 9:50
(E) At 10:00
6. How many more bricks are there in the larger stack?
(A) 4
(B) 5
(C) 6
(D) 7
Part B: Each correct answer is worth 4 points.
(E) 10
7. Ann has
. Barb gave Eve
. Jim has
. Bob has
. Who is Barb?
(A)
(B)
(C)
8. There is a path with square tiles.
(D)
(E)
How many tiles fit in the area inside?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9

Grade 1-2
Year 2013
9. Cat and Mouse are moving to the right. When Mouse jumps 1 tile, Cat jumps 2 tiles at the same time.
On which tile does Cat catch Mouse?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
10. I am a number. If you count by tens you will say my name. I am not ten. If you add me to 30, you will get a
number less than 60. Who am I?
(A) 20
(B) 30
(C) 40
(D) 50
(E) 60
11. There is a house on each corner of the streets. The houses
are shown on the map. Two new houses will be built on
each street between the corner houses. How many houses
will there be in all?
(A) 8
(B) 12
(C) 16
(D) 20
(E) Other answer
12. Kasia has 3 brothers and 3 sisters. How many brothers and how many sisters does her brother Mike have?
(A) 3 brothers and 3 sisters
(B) 3 brothers and 4 sisters
(C) 2 brothers and 3 sisters
(D) 3 brothers and 2 sisters
(E) 2 brothers and 4 sisters
Part C: Each correct answer is worth 5 points.
13. Ania makes a large cube from 27 small white cubes. She paints all the faces of the large cube. Then Ania removes four small cubes from four of the corners, as shown. While the paint is still wet, she stamps each of the new faces onto a piece of paper. How many of the following stamps can Ania make?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
14. Ann has a lot of these pieces:
She tries to put them in the square, as many as possible. How many cells shall be left empty?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

Grade 1-2
15. In a game it is possible to make the following exchanges:
Year 2013
Adam has 6 pears. How many strawberries will Adam have, when he trades all his pears for just
strawberries?
(A) 12
(B) 36
(C) 18
(D) 24
(E) 6
16. Sophie makes a row of 10 houses with matchsticks. In the picture you can see the beginning of the row. How many matchsticks does Sophie need altogether?
(A) 50
(B) 51
(C) 55
(D) 60
(E) 62
17. A square box is filled with two layers of identical square pieces of chocolate. Kirill has eaten all 20 pieces in the upper layer, which are along the walls of the box. How many pieces of chocolate are left in the box?
(A) 16
(B) 30
(C) 50
(D) 52
(E) 70
18. In a park there are babies in four-wheel strollers and children on two-wheel bikes. Paula counted wheels and the total was 12. When she added the number of strollers to the number of bikes, the total was 4. How many two-wheel bikes are there in the park?
(A) 1
(B) 2
(C) 3
(D) 4
(E) Other number

Grade 3-4
Year 2013
International Contest-Game MATH KANGAROO
Part A: Each correct answer is worth 3 points. 1. In which figure is the number of black kangaroos bigger than the number of white kangaroos?
(A)
(B)
(C)
(D)
(E)
2. Aline writes a correct calculation. Then she covers two digits which are the same with a sticker:
Which digit is under the stickers?
(A)
(B)
(C)
(D)
(E)
3. Monica arrived in the Kangaroo Camp on July 25th in the morning and left the camp on August 3rd in
the afternoon. How many nights did she sleep in the camp?
(A) 7
(B) 9
(C) 10
(D) 30
(E) 8
4. How many triangles of all sizes can be seen in the picture below?
(A) 9
(B) 10
(C) 11
(D) 13
(E) 12
5. In London 2012, the USA won the most medals: 46 gold, 29 silver and 29 bronze. China was second
with 38 gold, 27 silver and 23 bronze. How many more medals did the USA win compared to China?
(A) 6
(B) 14
(C) 16
(D) 24
(E) 26

Grade 3-4
Year 2013
6. There are three families in my neighbourhood with three children each; two of the families have
twins. All twins are boys. At most how many girls are in these families?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
7. Vero's mother prepares sandwiches with two slices of bread each. A package of bread has 24 slices.
How many sandwiches can she prepare from two and a half packages of bread?
(A) 24
(B) 30
(C) 48
(D) 34
(E) 26
8. About the number 325, five boys said:
Andrei: "This is a 3-digit number"
Boris: "All digits are distinct"
Vick: "The sum of the digits is 10"
Greg: "The units digit is 5"
Danny: "All digits are odd"
Which of the boys was wrong?
(A) Andrei
(B) Boris
(C) Vick
(D) Greg
(E) Danny
Part B: Each correct answer is worth 4 points. 9. The rectangular mirror was broken.
Which of the following pieces is the missing part of the broken mirror?
(A)
(B)
(C)
(D)
(E)
10. When Pinocchio lies, his nose gets 6 cm longer. When he tells the truth, his nose gets 2 cm shorter. When his nose was 9 cm long, he told three lies and made two true statements. How long was Pinocchio's nose afterwards?
(A) 14 cm
(B) 15 cm
(C) 19 cm
(D) 23 cm
(E) 31 cm

Grade 3-4
Year 2013
11. John is 33 years old. His three sons are 5, 6 and 10 years old. In how many years will the three sons together be as old as their father?
(A) 4
(B) 6
(C) 8
(D) 10
(E) 12
12. On the map, white lines represent streets. There are pictograms on some intersections (for example, trafic light, basket, tram). Ann started walking at the beginning of the middle vertical street in the direction of the arrow. At every intersection of streets she turned either to the right or to the left. First she turned right, then left, then again left, then right, then left, and finally again left. Which of the landmarks did Ann approach in the end?
(A)
(B)
(C)
(D)
(E)
13. Schoolmates Andy, Betty, Cathie and Dannie were born in the same year. Their birthdays were on February 20th, April 12th, May 12th and May 25th, not necessarily in this order. Betty and Andy were born in the same month. Andy and Cathie were born in the same day of different months. Who of these schoolmates is the oldest?
(A) Andy
(B) Betty
(C) Cathie (D) Dannie (E) impossible to determine
14. In the Adventure Park, 30 children took part in two of the adventures. 15 of them participated in the "moving bridge" contest, and 20 of them went down the zip-wire. How many of the children took part in both adventures?
(A) 25
(B) 15
(C) 30
(D) 10
(E) 5
15. Which of the five pieces in the answers fits with the piece in the separate picture, so that together they form a rectangle?
(A)
(B)
(C)
(D)
(E)
16. Children in the school club had to arrange fitness balls according to their size from the biggest to the smallest one. Rebecca was comparing them and said: the red ball is smaller than the blue one, the yellow one is bigger than the green one, and the green one is bigger than the blue one. What is the correct order of the fitness balls?
(A) green, yellow, blue, red (D) yellow, green, blue, red
(B) red, blue, yellow, green (E) blue, yellow, green, red
(C) yellow, green, red, blue

Grade 3-4
Year 2013
Part C: Each correct answer is worth 5 points.
17. In the shown triangle, first we join the midpoints of all the three sides. This way, we form a smaller triangle. We repeat this one more time with the smaller triangle, forming a new even smaller triangle, which we colour in red. How many triangles of the size of the red triangle are needed to cover completely the original triangle, without overlapping?
Note: Midpoint of a side is the point that divides the side in two parts of the same length.
(A) 5
(B) 8
(C) 10
(D) 16
(E) 32
18. There are oranges, apricots and peaches in a big basket. How many fruits are there in the basket if the peaches and the apricots together are 18, the oranges and the apricots together are 28 and 30 fruits are not apricots?
(A) 46
(B) 20
(C) 40
(D) 38
(E) 29
19. In December Tom-the-cat slept for exactly 3 weeks. Which calculations should we do in order to find how many minutes he stayed awake during this month?
(A) (31 – 7) × 3 × 24 × 60
(B) (31 – 7 × 3) × 24 × 60
(C) (30 – 7 × 3) × 24 × 60
(D) (31 – 7 ) × 24 × 60
(E) (31 – 7 × 3) × 24 × 60 × 60
20. Basil has several domino tiles, as shown in the figure. He wants to arrange them in a line according to the well-known "domino rule": in any two tiles that are next to each other, the squares that touch must have the same number of points. What is the largest number of tiles he can arrange in this way?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7
21. Cristi has to sell 10 glass bells that vary in price: 1 euro, 2 euro, 3 euro, 4 euro, 5 euro, 6 euro, 7 euro, 8 euro, 9 euro, 10 euro. In how many ways can Cristi divide all the glass bells in three packages so that all the packages have the same price?
(A) 1
(B) 2
(C) 3
(D) 4
(E) Such a division is not possible.

Grade 3-4
Year 2013
22. Nancy bought 17 cones of ice-cream for her three children. Misha ate twice as many cones as Ana. Dan ate more ice-cream than Ana but less than Misha. How many cones of ice-cream did Dan eat?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
23. Peter bought a carpet 36 dm wide and 60 dm long. The figure shows part of this carpet. As seen, the carpet has a pattern of small squares containing either a sun or a moon. You can count that along the width there are nine squares. When the carpet is fully unrolled, how many moons will be seen?
(A) 68
(B) 67
(C) 65
(D) 63
(E) 60
24. Beatrice has a lot of pieces like the grey one in the picture. At least how many of these grey pieces will she need to make a grey square?
(A) 3
(B) 4
(C) 6
(D) 8
(E) 16

Grade 11-12
International Contest-Game MATH KANGAROO
Part A: Each correct answer is worth 3 points.
Year 2013
1. Which of the following numbers is the largest?
(A) 2013
(B) 20+13
(C) 2013
(D) 2013
(E) 20 × 13
2. Four circles of radius 1 are touching each other and a smaller circle as seen in the picture. What is the radius of the smaller circle?
(A) 2 ?1
1 (B)
2
3 (C)
4
3 (D)
4
7 (E)
16
3. A three-dimensional object bounded only by polygons is called a polyhedron. What is the smallest
number of polygons that can bind a polyhedron, if we know that one of the polygons has 12 sides?
(A) 12
(B) 13
(C) 14
(D) 16
(E) 24
4. The cube root of 333 is equal to
(A) 33
(B) 333 ?1
(C) 323
(D) 332
(E) ( 3)3
5. The year 2013 has the property that its number is made up of the consecutive digits 0, 1, 2 and 3.
How many years have passed since the last time a year was made up of four consecutive digits?
(A) 467
(B) 527
(C) 581
(D) 693
(E) 990
6. Let f be a linear function for which f(2013) – f(2001) = 100. What is f(2031) – f(2013)?
(A) 75
(B) 100
(C) 120
(D) 150
(E) 180
7. Given that 2 < x < 3, how many of the following statements are true?
4 < x2 < 9
4 < 2x < 9
6 < 3x < 9 0 < x2 ? 2x < 3
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
8. Six superheroes capture 20 villains. The first superhero captures one villain, the second captures
two villains and the third captures three villains. The fourth superhero captures more villains than
any of the other five. What is the smallest number of villains the fourth superhero must have
captured?
(A) 7
(B) 6
(C) 5
(D) 4
(E) 3

Grade 11-12
Year 2013
9. In the cube to the right you see a solid, non-transparent pyramid ABCDS with base ABCD, whose vertex S lies exactly in the middle of an edge of the cube. You look at this pyramid from above, from below, from behind, from ahead, from the right and from the left. Which view does not arise?
(A)
(B)
(C)
(D)
(E)
10.
When
a certain
solid substance
melts,
its
volume increases
by
1 12
.
By how much does
its
volume
decrease when it solidifies again?
(A)
1 10
(B)
1 11
(C)
1 12
(D)
1 13
(E)
1 14
Part B: Each correct answer is worth 4 points.
11. The diagram shows two squares of equal side length placed so that
they overlap. The squares have a common vertex and the sides make an
angle of 45 degrees with each other, as shown. What is the area of the
overlap as a fraction of the area of one square?
1 (A)
2
1 (B)
2
(C) 1? 1 2
(D) 2 ?1
2 ?1 (E)
2
12.
How many positive integers n exist such that both
n 3
and 3n
are three-digit integers?
(A) 12
(B) 33
(C) 34
(D) 100
(E) 300
13. A circular carpet is placed on a floor of square tiles. All the tiles which have more than one point in common with the carpet are marked grey. Which of the following is an impossible outcome?
(A)
(B)
(C)
(D)
(E)
14. Consider the following statement about a function f on the set of integers: "For any even x, f(x) is even." What would be the negation of this proposition?
(A) For any even x, f(x) is odd
(B) For any odd x, f(x) is even
(C) For any odd x, f(x) is odd
(D) There exists an even number x such that f(x) is odd
(E) There exists an odd number x such that f(x) is odd

Grade 11-12
Year 2013
15. How many pairs (x,y) of positive integers satisfy the equation x2 y3 = 612 ?
(A) 6
(B) 8
(C) 10
(D) 12
(E) Another number.
16. Given a function W (x) = (a ? x)(b ? x)2 , where a < b. Its graph is in one of the following figures. In which one?
(A)
(B)
(C)
(D)
(E)
17. Consider a rectangle, one of whose sides has a length of 5. The rectangle can be cut into a square
and a rectangle, one of which has the area 4. How many such rectangles exist?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
18. Assume that x2 ? y2 = 84 , where x and y are positive integers. How many values may the
expression x2 + y2 have?
(A) 1
(B) 2
(C) 3
(D) 5
(E) 6
19. In the triangle ABC the points M and N on the side AB are such that AN = AC
and BM = BC. Find ∠ACB if ∠MCN = 43°.
(A) 86°
(B) 89°
(C) 90°
(D) 92°
(E) 94°
20. A box contains 900 cards numbered from 100 to 999. Any two cards have different numbers.
Fran?ois picks some cards and determines the sum of the digits on each of them. At least how many
cards must he pick in order to be certain to have three cards with the same sum?
(A) 51
(B) 52
(C) 53
(D) 54
(E) 55
Part C: Each correct answer is worth 5 points.
21. How many pairs (x,y) of integers with x ≤ y exist such that their product equals 5 times their sum?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
22. Let f (x), x ∈ R be the function defined by the following properties: f is periodic with period 5 and
f (x) = x2 when x ∈[?2,3) . What is f(2013) ?
(A) 0
(B) 1
(C) 2
(D) 4
(E) 9
23. We have many white cubes and many black cubes, all of the same size. We want to build a rectangular prism composed by exactly 2013 of these cubes so that they are placed alternating a white cube and a black cube in all directions. If we start putting a black cube in one of the eight corners of the prism, how many black squares will we see on the exterior surface of the solid?
(A) 887
(B) 888
(C) 890
(E) It depends on the dimensions of the prism
(D) 892

Grade 11-12
Year 2013
24. How many solutions (x,y), where x and y are real numbers, does the equation x2 + y2 = x + y
have? (A) 1
(B) 5
(C) 8
(D) 9
(E) Infinitely many.
25. There are 2013 points marked inside a square. Some of them are connected to the vertices of the
square and with each other so that the square is divided into non-overlapping triangles. All marked
points are vertices of these triangles. How many triangles are formed this way?
(A) 2013
(B) 2015
(C) 4026
(D) 4028
(E) impossible to determine
26. There are some straight lines drawn on the plane. Line a intersects exactly three other lines and line
b intersects exactly four other lines. Line c intersects exactly n other lines, with n ≠ 3, 4 .
Determine the number of lines drawn on the plane.
(A) 4
(B) 5
(C) 6
(D) 7
(E) Another number.
27. The sum of the first n positive integers is a three-digit number in which all of the digits are the
same. What is the sum of the digits of n?
(A) 6
(B) 9
(C) 12
(D) 15
(E) 18
28. On the island of Knights and Knaves there live only two types of people: Knights (who always speak the truth) and Knaves (who always lie). I met two men who lived there and asked the taller man if they were both Knights. He replied, but I could not figure out what they were, so I asked the shorter man if the taller was a Knight. He replied, and after that I knew which type they were. Were the men Knights or Knaves?
(A) They were both Knights.
(B) They were both Knaves.
(C) The taller was a Knight and the shorter was a Knave.
(D) The taller was a Knave and the shorter was a Knight.
(E) Not enough information is given.
29. Julian has written an algorithm in order to create a sequence of numbers as a1 = 1,
am+n = am + an + mn , where m and n are natural numbers. Find the value of a100.
(A) 100
(B) 1000
(C) 2012
(D) 4950
(E) 5050
30. The roundabout shown in the picture is entered by 5 cars at the same time, each
one from a different direction. Each of the cars drives less than one round and no
two cars leave the roundabout in the same direction. How many different
combinations are there for the cars leaving the roundabout?
(A) 24
(B) 44
(C) 60
(D) 81
(E) 120

Year 2013
Grade 1 and 2 DBACCB DEDABE DACBDB
Grade 3 and 4 DDBBCDBE BDBADEBD DDBCEBBB
Grade 5 and 6 ECCBEBBECD CCDBADDACD ADBABDBBDB
Grade 7 and 8 DBACEECEAC DEBCBAABBC AEDCCABDBC
Grade 9 and 10 DBCCBAECBC DBDADDBCEB DCCEEDCCBB
Grade 11 and 12 CABDCDEBED DAEDEADBEC ADCED*CBDEB
*Answer E was also accepted as correct for Q25
Answers