ĐỀ THI TOÁN QUỐC TẾ IMSO NĂM 2009 - Học tốt - Thích học toán

In the following grid, the area of the shaded region is … unit square.. Answer: 21 square units.[r]

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6th INTERNATIONAL MATHEMATICS AND SCIENCE OLYMPIAD FOR PRIMARY SCHOOL

IMSO 2009

SHORT ANSWER PROBLEMS

DIRECTORATE OF KINDERGARTEN AND PRIMARY DEVELOPMENT

DIRECTORATE GENERAL OF PRIMARY AND SECONDARY EDUCATION MANAGEMENT

MINISTRY OF NATIONAL EDUCATION Yogyakarta, 8 – 14 November 2009

1 A big square is formed from twenty five small squares Some of the small squares are black To make the big square symmetric about both diagonals, at least … additional small squares are needed to be colored black

Answer: 4

3 In a training program, an athlete must eat 154 eggs, during a period of time from November 8th till November 14th Every day in this period he must eat 6 more eggs than the previous day The number of eggs he eats on November 13th

is …

Solution: 34 eggs

Let a be the number of eggs eaten on November 8th Then 154=7a + (1+2+3+4+5+6)6=7a+126

So a=4 So the number of eggs eaten on November 13th is 4+5x6=34

Answer:

= 3 + 6 + 9 + 12 + … + 75

Answer: 4

5 In the following grid, the area of the shaded region is … unit square

Answer: 21 square units

6 Danny wants to create a set of cards of sizes from a sheet of paper of size The number of cards that can be made by Danny

is at most …

Answer: 23

7 Mrs Anna has 4 children: Alex, Brad, Christine, and Dennis Alex is not the youngest, but he is younger than Dennis If Brad’s age is the same as the mean of the ages of Alex and Dennis, then the oldest one is …

Solution: Dennis

Since Brad’s age is the same as the mean of ages of Alex and Dennis, then Brad is between Dennis and Alex Alex is not the youngest, so Christine is Thus the oldest is Dennis

8 In a math test, a correct answer will be marked 5 points and a wrong answers

points Tom answered all of the 35 questions and got a total score of 140 The number of questions Tom answered correctly is …

Answer: 30

# correct answer: 25 26 27 28 29 30 31 32

Total score: 105 112 119 126 133 140 147 154

9 The following shape is made from horizontal and vertical lines The lengths of some of the lines are given The perimeter of the shape is … unit

Answer : The perimeter is 2x((6+12)+(5+8))=62

10 Use numbers 2, 3, 4, 5, 7, and 8 exactly once to form two three-digit numbers P and Q If is a positive number; the smallest possible value of is

Answer: 36

P = 523

Q = 487 -

36

11 The natural numbers bigger than 1 are arranged in five columns as given by the following figure The number 2009 appears in the … column and the … row

Solution: 1st column, 502nd row

Observe that, the numbers appear on the first column are , with , where the rows occupied are the even rows, starting from 2 Since then 2009 appears on the first column and nd row

12 The integer 8 has two properties:

 If the number 1 is added, we get the number 9, which is a square number, i.e., 9 = 33

 Half of it is 4, which is also a square number, i.e., 4 = 22

The next natural number which has the same properties is …

The next natural number which has the same properties is …

Answer: 288

Number

(Even)

Propety 1 Property 2 Satisfy

8 8+1 = 9 = 3 2 8/2 = 4 = 2 2 1 and 2

24 24+1 = 25 = 5 2 24/2 = 12 1

48 48+1 = 49 = 7 2 48/2 = 24 1

80 80+1=81=9 2 80/2=40 1

120 120+1=121=11 2 120/6=60 1

168 168+1=169=13 2 168/2=84 1

224 224+1=225=15 2 224/2=112 1

288 288+1=289=17 2 288/2=144=12 2 1 and 2

The number = 288

A B

C

13 ABCD is a trapezoid (trapezium) with AB parallel to CD The ratio of AB : CD is

3 : 1 The point P is on CD The ratio of the area of triangle APB to the area of trapezoid ABCD is …:…

Answer: ¾ or 3 : 4

Wherever the position of P on CD, the ratio of

the areas of the triangle and the trapezoid is ¾

14 I have some marbles and some empty boxes If I try to put 9 marbles on each box, then there will be 2 empty boxes If I try to put 6 marbles in each box, then there will be 3 remaining marbles I have … boxes

Answer: 7 boxes

Case Box 1 Box 2 Box 3 … Box (n-2) Box (n-1) Box n Remaining

marbles

I 6 6 6 6 6 6 3

II 9 9 9 9 0 0 0

From case I, there should be 15 marbles that must be distributed into a multiple of 3 number of boxes

to get the case II Because 15 is a multiple of 3, then the only possible number of boxes that each has

9 marbles is 5 (and the other 2 boxes are empty) So the number of boxes are 5+2 = 7

15 The ten numbers 1,1, 2, 2, 3, 3, 4, 4, 5, 5, are arranged in a row (see the figure),

so that each number, except the first and last, is the sum or the difference of its two adjacent neighbors The value of X is

2

The ten numbers 1,1,2,2,3,3,4,4,5,5, are arranged in a row (see figure), so that each number, except the first and last, is the sum or the difference of its two adjacent neighbors The value of X is Answer: 4,1,5,4,1,3,2,5,3,2 So, X=3

16 In a football competition, if a team wins it will get 3 points If it draws it will get 1 point, and if it loses it will get 0 points After playing 20 times, Team B gets the total score of 53 Team B loses at least … times

Answer:1

# Wins 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 Win-Score 60 57 54 51 48 45 42 39 36 33 30 27 24 21 18

# Loses 0 1 0

Draw-Score 0 2 4 5 6 7 8

# Scores 60 53 52 50 48

17 Among 8 points located in a plane, five of them lie on one line Any three points are selected from those 8 points as corner points of a triangle There are at most … triangles that can be formed

Answer: 46

There are several possibilities triangle formed

 Triangle formed by points outside the line In this case there is exactly one triangle

 Triangle formed by a point outside the line and two points on the line There are 10 pairs of

points from 5 points located on the line Since there are 3 points out of line, then the number

of triangles that might be formed is 3 x 10 = 30 triangles

 Triangle formed by a point on the line and two points outside the line There are 3 pairs of

points from beyond the 3 point line Since there are 5 points on the line, there is a 5 x 3 = 15 triangle that may be formed

 So, overall there is (1 + 30 + 15 = 46) triangles that may be formed

18 Fill in all the numbers 0,1,2,3,4,5,6,7,8,9 on the ten squares below, so that the sum of numbers located on each arrowed line is 20 Two numbers are already filled in

The number on the square with a question mark ("?") is

Answer: 7

2.? + 2.8 + 2 +5 +6 +9 +3 +1 +4 =60

2.? + 46 = 60

? = 7

One of the suitable arrangements is given in the following figure:

19 Nine dots are arranged as shown in the figure below, where ABCD is a square AT=TD, DS=SC, CR=BR, and AP=PQ=QB A triangle can be constructed by lining from dot to dot At most, there are … different right triangles that one can construct so that at least one of the dots P,Q,R,S, and T is its vertex

Answer: 21

From dot P, triangles PBR,PBC, PAT, PAD (4)

From dot Q, triangles QBR, QBC, QAT, QAD (4)

From dot R, triangles RCS,RCD,RCT, RST, RDT, RTA,RAB,RBT (8)

0

8

1

4

20

20

20

20 ABCDE is a five-digit positive number ABCDE1 is three times 1ABCDE ABCDE is …

Answer: 42857

Let x = ABCDE

3(100000 + x) = 10x+1

10x+1 = 3(100000 + x)

10x+1 = 300000 + 3x

10x = 299999 + 3x

7x = 299999

x= 299999/7 = 42857

21 In the diagram below, BC=5, DE=1 and DC=20, where D lies on AC and E lies

on AB Both ED and BC are perpendicular to AC The length of AD is …

(Note: the figure is not in proportional scale)

Answer : 5

22 The number N consists of three different digits and is greater than 200 The digits are greater than 1 For any two digits, one digit is a multiple of the other or the difference is 3 For example, 258 is one of such numbers There are at most … possible N

Answer: 18 numbers

Possibilities: 248;284;428;482;824;842  6 numbers

258;285;528;582;825;852  6 numbers

369;396; 639;693;936;963  6 numbers

- +

18 numbers

23 In the figure, two half-circles are inscribed in a square These two half-circles intersect at the center of the square If the side of the square has length 14 cm, then the area of the shaded region is … cm2

Answer: 56 OR ½(2π-1)49 OR 55.86

See the picture on the side

The shaded area consists of four parts which are congruent

The area I = area of a quarter circle - area of triangle

=772 492

=14

24 Replace the asterisks with digits so that the multiplication below is correct:

The product is …

Answer: 182720097

3 3 3 3 7

A B C 1

3 3 3 3 7

* * * * * D

* * * * * *

* * * * * *

* * * * 2 0 0 9 7

III

IV

3 3 3 3 7

A B 8 1

3 3 3 3 7

2 6 6 6 9 6

* * * * * E

* * * * * *

* * * * 2 0 0 9 7

3 3 3 3 7

A B 8 1

3 3 3 3 7

2 6 6 6 9 6

1 3 3 3 4 8

* * * * * F

* * * * 2 0 0 9 7

3 3 3 3 7

A B 8 1

3 3 3 3 7

2 6 6 6 9 6

1 3 3 3 4 8

1 6 6 6 8 5

1 8 2 7 2 0 0 9 7

25 The areas of 3 sides of a block are 44 cm2, 33 cm2, and 48 cm2 The volume of the block is … cm3

Answer : 264 cm 3

Method 1

If the size of the edges block row is 4 cm, 3 cm and 11 cm, the area of the front and side would be appropriate, ie, respectively 44 and 33 However, these measures are not suitable for the wide side.

Because 11 is the Greatest Common Divisor of 33 and 44, so we can modify the measures so that the vertices are in accordance with the broad sides of the unknown.

The size of the vertices block row are 8 cm, 6 cm, and 2

11

Thus, the block volume is 8 x 6 x

2

11

= 264 cm3

Method 2

For example the size of the vertices block is x, y, and z as shown in the following figure.

Can be written: xy = 48, xz = 44, and yz = 33

Retrieved

(xy) x (xz) x (yz) = 48 x 44 x 33 = (4 x 12) x (4 x 11) x (3 x 11)

or

2 2

2y z

x = (4 x 4 ) x (11 x 11) x (12 x 3) = (4 x 4 ) x (11 x 11) x (6 x 6)

or

xyz = 4 x 11 x 6 = 264

Thus, the block volume is 264 cm 3

33

48

4x2

2 11

3x2

44

33

48

x

y

z

44

33

12?

4

11

3

44