Time : 50 minutes Passing marks : 70%
Answer : End of the problem
Problem 1
(1)
(29/25 + 1.2 × 2.45) / (7.875 × 8/21 - 0.95) =
(2)
A number is added in an order from 1 as 1 + 2 + 3 + ------.
Problem 3
Problem 4
In a certain zoo, rabbits are kept in the hut which looks like the form as shown in Fig. 1 when it is seen from right above.
Problem 5
When a certain number was added, the sum exceeded 1000 for the first time.
Find both of the number and the sum.
(3)
As shown in a figure, an integer is arranged sequentially from 1.
Problem 2
The case where 2007 steps are put in order is considered.
(1) Find the largest number.
(2) In which steps from the top and which number from the left is the largest number ?
Problem 2
As shown in a figure, eight cubes of 1 cm3 in volume are accumulated.
When this solid is cut by the plane which passes along the side AB and CD, find the volume of the solid of the smaller one.
Problem 3
As shown in Fig. 1, the circumference is equally divided into five, and a star shape is made.
This circle and star shape are set as shown in Fig. 2.
First, fixing the point E to the circumference, the star shape is clockwise rotated until the point D attaches to the circumference.
Next, fixing the point D to the circumference, the star shape is clockwise rotated until the point C attaches to the circumference.
This motion is repeated until the star shape returns to the original state of Fig. 2 again.
Draw the locus while the point A moves in this time by using compass.
Problem 4
In the inside of the hut, there are transparent boards at the position of the dotted line in the figure and it is divided into eight rooms.
The shadow portion in the middle is not a room.
There are four windows in the hut and we can see rabbits by looking through the windows.
For example, when rabbits are kept as shown in Fig. 2, four rabbits are visible from any window and there are nine in total.
(1) There may be a room in which no rabbit is.
Rabbits should be kept to be seen nine from any window.
In this case, how many rabbits are there at least and how many rabbits at most?
(2) There is at least one rabbit kept in every room and 17 rabbits in total are kept in the hut.
In order for 7 rabbits to be seen from any window, how many rabbits should be kept in each room?
Answer one pattern in the figure.
For example, when putting in a rabbit as shown in Fig. 2, it should be answered as shown in Fig. 3.
For example, when putting in a rabbit as shown in Fig. 2, it should be answered as shown in Fig. 3.
Problem 5
We are going to make some origami cranes by some sheets of origami (paper folding).
Problem 6
In order to fold all cranes, the time does not change even if it is done by five persons group or six persons group.
However if it is done by seven persons party, the time will be shorter.
How many sheets of origami is there in all?
How many sheets of origami is there in all?
Answer all the possible number of sheets considered.
Noted that the time for one crane to be folded does not change by people.
Problem 6
Some rectangular pasteboard is cut off along a diagonal line, as shown in Fig. 1.
By using this I would like to make the glass which has the form of the triangular pyramid as shown in Fig. 2.
However, it may be unable to make or able to make depending on the length of the side AB.
Answer the conditions of the length of the side AB in the case of being able to make.
However, it may be unable to make or able to make depending on the length of the side AB.
Answer the conditions of the length of the side AB in the case of being able to make.
<Answer>
Problem 1
(1)
(29/25 + 1.2 × 2.45) / (7.875 × 8/21 - 0.95) =
Answer
2
(2)
A number is added in an order from 1 as 1 + 2 + 3 + ------.
When a certain number was added, the sum exceeded 1000 for the first time.
Find both of the number and the sum.
Answer
Problem 3
Problem 4
In a certain zoo, rabbits are kept in the hut which looks like the form as shown in Fig. 1 when it is seen from right above.
Number is 45,
Sum is 1035
Solution
1+2+3+・・・・・・・・+40 = (1+40) × 40 / 2 = 820
820+41+42+43+44=990
990+45=1035
(3)
As shown in a figure, an integer is arranged sequentially from 1.
The case where 2007 steps are put in order is considered.
(1) Find the largest number.
(2) In which steps from the top and which number from the left is the largest number ?
Answer
Problem 2
(1) 2015028
(2) 1338th from the top, 670th from the left
(2) 1338th from the top, 670th from the left
Solution
(1)
1+2+3+・・・・・・+2007= (1+2007) × 2007 / 2 = 2015028
(2)
2007 / 3 = 669 → same as six steps pattern
2007 - 669 = 1338th
669 + 1 = 670th
Problem 2
As shown in a figure, eight cubes of 1 cm3 in volume are accumulated.
When this solid is cut by the plane which passes along the side AB and CD, find the volume of the solid of the smaller one.
Answer
29/12 cm3
Solution
Fig.2 = 3 × 2 / 2 - 1 = 2 cm2
△EFG = 1/3 × 1/2 × 1/2 = 1/12 cm2
△IDH = 2/3 × 1 × 1/2 = 1/3 cm2
Total volume = (2+ 1/12 + 1/3) × 1 = 29/12 cm3
Problem 3
As shown in Fig. 1, the circumference is equally divided into five, and a star shape is made.
This circle and star shape are set as shown in Fig. 2.
First, fixing the point E to the circumference, the star shape is clockwise rotated until the point D attaches to the circumference.
Next, fixing the point D to the circumference, the star shape is clockwise rotated until the point C attaches to the circumference.
This motion is repeated until the star shape returns to the original state of Fig. 2 again.
Draw the locus while the point A moves in this time by using compass.
Answer
180 × (5 - 2 ) = 108 degrees
(180 - 108) / 2 = 36 degrees
∠DED´= 360 - 108 × 2 = 144 degrees
Central angle of each arc is 144 degrees.
Problem 4
In the inside of the hut, there are transparent boards at the position of the dotted line in the figure and it is divided into eight rooms.
The shadow portion in the middle is not a room.
There are four windows in the hut and we can see rabbits by looking through the windows.
For example, when rabbits are kept as shown in Fig. 2, four rabbits are visible from any window and there are nine in total.
(1) There may be a room in which no rabbit is.
Rabbits should be kept to be seen nine from any window.
In this case, how many rabbits are there at least and how many rabbits at most?
(2) There is at least one rabbit kept in every room and 17 rabbits in total are kept in the hut.
In order for 7 rabbits to be seen from any window, how many rabbits should be kept in each room?
Answer one pattern in the figure.
For example, when putting in a rabbit as shown in Fig. 2, it should be answered as shown in Fig. 3.
For example, when putting in a rabbit as shown in Fig. 2, it should be answered as shown in Fig. 3.
Answer
Problem 5
(1) 18 rabbits at least, 36 at most
(2) Examples of answers
(2) Examples of answers
Solution
(1)
Left side is least and right side is most.
(2)
Problem 5
We are going to make some origami cranes by some sheets of origami (paper folding).
In order to fold all cranes, the time does not change even if it is done by five persons group or six persons group.
However if it is done by seven persons party, the time will be shorter.
How many sheets of origami is there in all?
How many sheets of origami is there in all?
Answer all the possible number of sheets considered.
Noted that the time for one crane to be folded does not change by people.
Answer
Problem 6
7, 13, 14, 19, 20, 25
Solution
| Sheet | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 |
| 5 persons | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 |
| 6 p. | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 |
| 7 p. | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 |
| 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |
| 5 | 5 | 6 | 6 | 6 | 6 | 6 | 7 |
| 4 | 5 | 5 | 5 | 5 | 5 | 5 | 6 |
| 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 |
Problem 6
Some rectangular pasteboard is cut off along a diagonal line, as shown in Fig. 1.
By using this I would like to make the glass which has the form of the triangular pyramid as shown in Fig. 2.
However, it may be unable to make or able to make depending on the length of the side AB.
Answer the conditions of the length of the side AB in the case of being able to make.
However, it may be unable to make or able to make depending on the length of the side AB.
Answer the conditions of the length of the side AB in the case of being able to make.
Answer
AB > AE
Solution
∠AEB + ∠CED > ∠AED
∠AEB × 2 > ∠AED
∠AEB + ∠AED = 180 degrees
∠AEB = 180 / 3 = 60 degrees
∠AEB should be larger than 60 degrees.
When ∠AEB = 60 degrees, AB = AE.
When ∠AEB > 60 degrees, AB > AE.
