IMSO 2009
SHORT ANSWER PROBLEMS
DIRECTORATE OF KINDERGARTEN AND PRIMARY DEVELOPMENT
DIRECTORATE GENERAL OF PRIMARY AND SECONDARY EDUCATION MANAGEMENT MINISTRY OF NATIONAL EDUCATION
Short Answer Yogyakarta, 8-14 November
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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.
2. If , then =….
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.
4. ….
Answer:
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= 3 + 6 + 9 + 12 + … + 75
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
Short Answer Yogyakarta, 8-14 November
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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
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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.
I II III IV V
2 3 4 5
9 8 7 6
10 11 12 13
17 16 15 14
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 = 33.
Half of it is 4, which is also a square number, i.e., 4 = 22. 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 = 32 8/2 = 4 = 22 1 and 2 24 24+1 = 25 = 52 24/2 = 12 1 48 48+1 = 49 = 72 48/2 = 24 1
80 80+1=81=92 80/2=40 1
120 120+1=121=112 120/6=60 1
Short Answer Yogyakarta, 8-14 November
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 ... .
4 1 X 3
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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
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
Short Answer Yogyakarta, 8-14 November
Page 8 of 12 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)
7 2 5 6
0 8
1
4
9 3
20
20
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From dot S, triangles SDT,SDA,SCB (3) From dot T, triangles TDC,TAB (2)
Total:
21
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
Short Answer Yogyakarta, 8-14 November
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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: 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
Page 11 of 12 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
For example the size of the vertices block is x, y, and z as shown in the following figure.
Short Answer Yogyakarta, 8-14 November
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or
xyz = 4 x 11 x 6 = 264.
